November  2014, 19(9): 2889-2913. doi: 10.3934/dcdsb.2014.19.2889

Stochastically perturbed sliding motion in piecewise-smooth systems

1. 

Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand

2. 

Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

Received  April 2013 Revised  February 2014 Published  September 2014

Sliding motion is evolution on a switching manifold of a discontinuous, piecewise-smooth system of ordinary differential equations. In this paper we quantitatively study the effects of small-amplitude, additive, white Gaussian noise on stable sliding motion. For equations that are static in directions parallel to the switching manifold, the distance of orbits from the switching manifold approaches a quasi-steady-state density. From this density we calculate the mean and variance for the near sliding solution. Numerical results of a relay control system reveal that the noise may significantly affect the period and amplitude of periodic solutions with sliding segments.
Citation: D. J. W. Simpson, R. Kuske. Stochastically perturbed sliding motion in piecewise-smooth systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2889-2913. doi: 10.3934/dcdsb.2014.19.2889
References:
[1]

A. J. Van der Schaft and J. M. Schumacher, An Introduction to Hybrid Dynamical Systems,, Springer-Verlag, (2000).

[2]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, volume 18 of Lecture Notes in Applied and Computational Mathematics,, Springer-Verlag, (2004). doi: 10.1007/978-3-540-44398-8.

[3]

S. Banerjee and G. C. Verghese, Nonlinear Phenomena in Power Electronics,, IEEE Press, (2001). doi: 10.1109/9780470545393.

[4]

Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems,, World Scientific, (2003).

[5]

T. Puu and I. Sushko, editors, Business Cycle Dynamics: Models and Tools,, Springer-Verlag, (2006).

[6]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications,, Springer-Verlag, (2008).

[7]

N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems,, Springer, (2006).

[8]

B. Lindner, J. Garcia-Ojalvo, A. Neiman and L. Schimansky-Geier, Effects of noise in excitable systems,, Phys. Reports, 392 (2004), 321. doi: 10.1016/j.physrep.2003.10.015.

[9]

A. S. Pikovsky and J. Kurths, Coherence resonance in a noise-driven excitable system,, Phys. Rev. Lett., 78 (1997), 775. doi: 10.1103/PhysRevLett.78.775.

[10]

L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic resonance,, Rev. Modern Phys., 70 (1998), 223. doi: 10.1103/RevModPhys.70.223.

[11]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Kluwer Academic Publishers., (1988). doi: 10.1007/978-94-015-7793-9.

[12]

M. Wiercigroch and B. De Kraker, Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities,, Singapore, (2000). doi: 10.1142/9789812796301.

[13]

B. Brogliato, Nonsmooth Mechanics: Models, Dynamics and Control,, Springer-Verlag, (1999).

[14]

B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak and J. Wojewoda, Chaotic Mechanics in Systems with Impacts and Friction,, World Scientific, (1999). doi: 10.1142/9789812798565.

[15]

J. Awrejcewicz and C. Lamarque, Bifurcation and Chaos in Nonsmooth Mechanical Systems,, World Scientific, (2003). doi: 10.1142/9789812564801.

[16]

R. A. Ibrahim, Vibro-Impact Dynamics., volume 43 of Lecture Notes in Applied and Computational Mechanics,, Springer, (2009). doi: 10.1007/978-3-642-00275-5.

[17]

C. K. Tse, Complex Behavior of Switching Power Converters,, CRC Press, (2003). doi: 10.1109/JPROC.2002.1015006.

[18]

A. F. Filippov, Differential equations with discontinuous right-hand side., Mat. Sb., 51 (1960), 99.

[19]

P. Casini, O. Giannini and F. Vestroni, Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics,, Nonlinear Dyn., 46 (2006), 259. doi: 10.1007/s11071-006-9041-0.

[20]

A. C. J. Luo and B. C. Gegg, Stick and non-stick periodic motions in periodically forced oscillators with dry friction,, J. Sound Vib., 291 (2006), 132. doi: 10.1016/j.jsv.2005.06.003.

[21]

M. Johansson, Piecewise Linear Control Systems., volume 284 of Lecture Notes in Control and Information Sciencesm, Springer-Verlag, (2003). doi: 10.1007/3-540-36801-9.

[22]

M. di Bernardo, K. H. Johansson and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations,, Int J. Bifurcation Chaos, 11 (2001), 1121. doi: 10.1142/S0218127401002584.

[23]

Z. Schuss, Theory and Applications of Stochastic Differential Equations,, Wiley, (1980).

[24]

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,, Springer-Verlag, (1985).

[25]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems,, Springer, (2012). doi: 10.1007/978-3-642-25847-3.

[26]

J. Grasman and O. A. van Herwaarden, Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications,, Springer, (1999). doi: 10.1007/978-3-662-03857-4.

[27]

, Special Issue on Nonsmooth Systems,, Phys. D, 241 (2012).

[28]

M. Dutta, H. E. Nusse, E. Ott, J. A. Yorke and G. Yuan, Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems,, Phys. Rev. Lett., 83 (1999), 4281. doi: 10.1103/PhysRevLett.83.4281.

[29]

T. C. L. Griffin, Dynamics of Stochastic Nonsmooth Systems,, PhD thesis, (2005).

[30]

T. Griffin and S. Hogan, Dynamics of discontinuous systems with imperfections and noise,, in IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, (2005), 275. doi: 10.1007/1-4020-3268-4_26.

[31]

P. Glendinning, The border collision normal form with stochastic switching surface,, SIAM J. Appl. Dyn. Sys., 13 (2014), 181. doi: 10.1137/130931643.

[32]

M. A. Hassouneh, E. H. Abed and H. E. Nusse, Robust dangerous border-collision bifurcations in piecewise smooth systems,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.070201.

[33]

A. Ganguli and S. Banerjee, Dangerous bifurcation at border collision: When does it occur?,, Phys. Rev. E., 71 (2005). doi: 10.1103/PhysRevE.71.057202.

[34]

Y. Do, A mechanism for dangerous border collision bifurcations,, Chaos Solitons Fractals, 32 (2007), 352. doi: 10.1016/j.chaos.2006.07.018.

[35]

Y. Do and H. K. Baek, Dangerous border-collision bifurcations of a piecewise-smooth map,, Comm. Pure Appl. Anal., 5 (2006), 493. doi: 10.3934/cpaa.2006.5.493.

[36]

R. Wackerbauer, Noise-induced stabilization of one-dimensional discontinuous maps,, Phys. Rev. E, 58 (1998), 3036.

[37]

L. Zhang, P. Shi, C. Wang and H. Gao, Robust $H_\infty$ filtering for switched linear discrete-time systems with polytopic uncertainties,, Int. J. Adapt. Control Signal Process, 20 (2006), 291. doi: 10.1002/acs.901.

[38]

W. Zhang, J. Hu and J. Lian, Quadratic optimal control of switched linear stochastic systems,, Syst. Contr. Lett., 59 (2010), 736. doi: 10.1016/j.sysconle.2010.08.010.

[39]

P. H. E. Tiesinga, Precision and reliability of periodically and quasiperiodically driven integrate-and-fire neurons,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.041913.

[40]

A. B. Nordmark, Non-periodic motion caused by grazing incidence in impact oscillators,, J. Sound Vib., 2 (1991), 279. doi: 10.1016/0022-460X(91)90592-8.

[41]

H. Dankowicz and A. B. Nordmark, On the origin and bifurcations of stick-slip oscillations,, Phys. D, 136 (2000), 280. doi: 10.1016/S0167-2789(99)00161-X.

[42]

M. H. Fredriksson and A. B. Nordmark, On normal form calculation in impact oscillators,, Proc. R. Soc. A, 456 (2000), 315. doi: 10.1098/rspa.2000.0519.

[43]

M. di Bernardo, C. J. Budd and A. R. Champneys, Normal form maps for grazing bifurcations in $n$-dimensional piecewise-smooth dynamical systems,, Phys. D, 160 (2001), 222. doi: 10.1016/S0167-2789(01)00349-9.

[44]

D. J. W. Simpson, J. Hogan and R. Kuske, Stochastic regular grazing bifurcations,, SIAM J. Appl. Dyn. Sys., 12 (2013), 533. doi: 10.1137/120884286.

[45]

M. F. Dimentberg and D. V. Iourtchenko, Random vibrations with impacts: A review,, Nonlinear Dyn., 36 (2004), 229. doi: 10.1023/B:NODY.0000045510.93602.ca.

[46]

M. F. Dimentberg and A. I. Menyailov, Response of a single-mass vibroimpact system to white-noise random excitation,, Z. Angew. Math. Mech., 59 (1979), 709. doi: 10.1002/zamm.19790591205.

[47]

N. Sri Namachchivaya and J. H. Park, Stochastic dynamics of impact oscillators,, J. Appl. Mech. Trans. ASME, 72 (2005), 862. doi: 10.1115/1.2041660.

[48]

J. Feng, W. Xu, H. Rong and R. Wang, Stochastic responses of Duffing-Van der Pol vibro-impact system under additive and multiplicative random excitations,, Int. J. Non-Linear Mech., 44 (2009), 51. doi: 10.1016/j.ijnonlinmec.2008.08.013.

[49]

V. F. Zhuravlev, A method for analyzing vibration-impact systems by means of special functions,, Mech. Solids, 11 (1976), 23.

[50]

P. D. Christofides and N. H. El-Farra, Control of Nonlinear and Hybrid Process Systems. Designs for Uncertainty, Constraints and Time-Delays,, Springer, (2005).

[51]

J. Raouf and H. Michalska, Robust stabilization of switched linear systems with Wiener process noise,, in 49th IEEE Conference on Decision and Control, (2010), 6493. doi: 10.1109/CDC.2010.5717694.

[52]

W. Feng and J.-F. Zhang, Stability analysis and stabilization control of multi-variable switched stochastic systems,, Automatica, 42 (2006), 169. doi: 10.1016/j.automatica.2005.08.016.

[53]

D. Chatterjee and D. Liberzon, On stability of stochastic switched systems,, in Proceedings of the 43rd IEEE Conference on Decision and Control., (2004), 4125. doi: 10.1109/CDC.2004.1429398.

[54]

E. Skafidas, R. J. Evans, A. V. Savkin and I. R. Petersen, Stability results for switched controller systems,, Automatica, 35 (1999), 553. doi: 10.1016/S0005-1098(98)00167-8.

[55]

P. Mhaskar, N. H. El-Farra and P. D. Christofides, Robust hybrid predictive control of nonlinear systems,, Automatica, 41 (2005), 209. doi: 10.1016/j.automatica.2004.08.020.

[56]

B. Hu, X. Xu, P.J. Antsaklis and A. N. Michel, Robust stabilizing control laws for a class of second-order switched systems,, Syst. Control Lett., 38 (1999), 197. doi: 10.1016/S0167-6911(99)00065-1.

[57]

Z. Sun, A robust stabilizing law for switched linear systems,, Int. J. Control, 77 (2004), 389. doi: 10.1080/00207170410001667468.

[58]

D. Liberzon, Switching in Systems and Control,, Birkhauser, (2003). doi: 10.1007/978-1-4612-0017-8.

[59]

S.-C. Tan, Y.-M. Lai and C. K. Tse, Sliding Mode Control of Switching Power Converters,, CRC Press, (2012).

[60]

J.-Q. Sun, Stochastic Dynamics and Control., volume 4 of Nonlinear Science and Complexity,, Elsevier, (2006).

[61]

Y. Niu, D. W. C. Ho and J. Lam, Robust integral sliding mode control for uncertain stochastic systems with time-varying delay,, Automatica, 41 (2005), 873. doi: 10.1016/j.automatica.2004.11.035.

[62]

L. Wu, D. W. C. Ho and C. W. Li, Stabilisation and performance synthesis for switched stochastic systems,, IET Control Theory Appl., 4 (2010), 1877. doi: 10.1049/iet-cta.2009.0179.

[63]

L. Wu, D. W. C. Ho and C. W. Li, Sliding mode control of switching hybrid systems with stochastic perturbation,, Syst. Contr. Lett., 60 (2011), 531. doi: 10.1016/j.sysconle.2011.04.007.

[64]

Y. Niu, D. W. C. Ho and X. Wang, Robust $H_\infty$ control for nonlinear stochastic systems: A sliding-mode approach,, IEEE Trans. Automat. Contr., 53 (2008), 1695. doi: 10.1109/TAC.2008.929376.

[65]

D. E. Stewart, Rigid-body dynamics with friction and impact,, SIAM Rev., 42 (2000), 3. doi: 10.1137/S0036144599360110.

[66]

M. Abadie, Dynamical simulation of rigid bodies: Modelling of frictional contact,, in Impacts in Mechanical Systems: Analysis and Modelling, (2000). doi: 10.1007/3-540-45501-9_2.

[67]

P. S. Goohpattader, S. Mettu and M. K. Chaudhury, Experimental investigation of the drift and diffusion of small objects on a surface subjected to a bias and an external white noise: Roles of Coulombic friction and hysteresis,, Langmuir, 25 (2009), 9969. doi: 10.1021/la901111u.

[68]

P. S. Goohpattader and M. K. Chaudhury, Diffusive motion with nonlinear friction: Apparently Brownian,, J. Chem. Phys., 133 (2010). doi: 10.1063/1.3460530.

[69]

P.-G. de Gennes, Brownian motion with dry friction,, J. Stat. Phys., 119 (2005), 953. doi: 10.1007/s10955-005-4650-4.

[70]

H. Hayakawa, Langevin equation with Coulomb friction,, Phys. D, 205 (2005), 48. doi: 10.1016/j.physd.2004.12.011.

[71]

H. Touchette, E. Van der Straeten and W. Just, Brownian motion with dry friction: Fokker-Planck approach,, J. Phys. A: Math. Theor., 43 (2010). doi: 10.1088/1751-8113/43/44/445002.

[72]

A. Baule, E. G. D. Cohen and H. Touchette, A path integral approach to random motion with nonlinear friction,, J. Phys. A: Math. Theor., 43 (2010). doi: 10.1088/1751-8113/43/2/025003.

[73]

A. Baule, H. Touchette and E. G. D. Cohen, Stick-slip motion of solids with dry friction subject to random vibrations and an external field,, Nonlinearity, 24 (2011), 351. doi: 10.1088/0951-7715/24/2/001.

[74]

H. Touchette, T. Prellberg and W. Just, Exact power spectra of Brownian motion with solid friction,, J. Phys. A: Math. Theor., 45 (2012). doi: 10.1088/1751-8113/45/39/395002.

[75]

D. J. W. Simpson and R. Kuske, Stochastic perturbations of periodic orbits with sliding, Submitted to: J. Nonlin. Sci., (2014).

[76]

A. Colombo, M. di Bernardo, S. J. Hogan and M. R. Jeffrey, Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems,, Phys. D, 241 (2012), 1845. doi: 10.1016/j.physd.2011.09.017.

[77]

M. di Bernardo, P. Kowalczyk and A. Nordmark, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings,, Phys. D, 170 (2002), 175. doi: 10.1016/S0167-2789(02)00547-X.

[78]

A. Colombo and M. R. Jeffrey, Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows,, SIAM J. Appl. Dyn. Sys., 10 (2011), 423. doi: 10.1137/100801846.

[79]

M. R. Jeffrey and A. Colombo, The two-fold singularity of discontinuous vector fields,, SIAM J. Appl. Dyn. Sys., 8 (2009), 624. doi: 10.1137/08073113X.

[80]

M. A. Teixeira, Stability conditions for discontinuous vector fields,, J. Differential Equations, 88 (1990), 15. doi: 10.1016/0022-0396(90)90106-Y.

[81]

T.-S. Chiang and S.-J. Sheu, Large deviation of small perturbation of some unstable systems,, Stoch. Anal. Appl., 15 (1997), 31. doi: 10.1080/07362999708809462.

[82]

R. Kuske and G. Papanicolaou, The invariant density of a chaotic dynamical system with small noise,, Phys. D, 120 (1998), 255. doi: 10.1016/S0167-2789(98)00085-2.

[83]

P. Hitczenko and G. S. Medvedev, Bursting oscillations induced by small noise,, SIAM J. Appl. Math., 69 (2009), 1359. doi: 10.1137/070711803.

[84]

Ya. Z. Tsypkin, Relay Control Systems,, Cambridge University Press, (1984).

[85]

G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems,, Prentice Hall, (2002).

[86]

R. C. Dorf and R. H. Bishop, Modern Control Systems,, Prentice Hall, (2001). doi: 10.1109/TSMC.1981.4308749.

[87]

K. J. Åström and R. M. Murray, Feedback Systems. An Introduction for Scientists and Engineers,, Princeton University Press, (2008).

[88]

K. H. Johansson, A. Rantzer and K. J. Åström, Fast switches in relay feedback systems,, Automatica, 35 (1999), 539. doi: 10.1016/S0005-1098(98)00160-5.

[89]

K. H. Johansson, A. E. Barabanov and K. J. Åström, Limit cycles with chattering in relay feedback systems,, IEEE Trans. Automat. Contr., 47 (2002), 1414. doi: 10.1109/TAC.2002.802770.

[90]

Y. Zhao, J. Feng and C. K. Tse, Discrete-time modeling and stability analysis of periodic orbits with sliding for switched linear systems,, IEEE Trans. Circuits Systems I Fund. Theory Appl., 57 (2010), 2948. doi: 10.1109/TCSI.2010.2050230.

[91]

F. Dercole, R. Ferrière, A. Gragnani and S. Rinaldi, Coevolution of slow-fast populations: Evolutionary sliding, evolutionary pseudo-equilibria and complex Red Queen dynamics,, Proc. R. Soc. B, 273 (2006), 983. doi: 10.1098/rspb.2005.3398.

[92]

F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models,, Theor. Popul. Biol., 72 (2007), 197. doi: 10.1016/j.tpb.2007.06.003.

[93]

J. A. Amador, G. Olivar and F. Angulo, Smooth and Filippov models of sustainable development: Bifurcations and numerical computations,, Differ. Equ. Dyn. Syst., 21 (2013), 173. doi: 10.1007/s12591-012-0138-2.

[94]

S. Tang, J. Liang, Y. Xiao and R. A. Cheke, Sliding bifurcations of Filippov two stage pest control models with economic thresholds,, SIAM J. Appl. Math., 72 (2012), 1061. doi: 10.1137/110847020.

[95]

R. Szalai and H. M. Osinga, Invariant polygons in systems with grazing-sliding,, Chaos, 18 (2008). doi: 10.1063/1.2904774.

[96]

M. Tanelli, G. Osorio, M. di Bernardo, S. M. Savaresi and A. Astolfi, Existence, stability and robustness analysis of limit cycles in hybrid anti-lock braking systems,, Int. J. Contr., 82 (2009), 659. doi: 10.1080/00207170802203598.

[97]

M. di Bernardo, K. H. Johansson, U. Jönsson and F. Vasca, On the robustness of periodic solutions in relay feedback systems,, in 15th Triennial World Congress, (2002).

[98]

F. Flandoli, Random Perturbations of PDEs and Fluid Dynamic Models., volume 2015 of Lecture Notes in Mathematics,, Springer, (2011). doi: 10.1007/978-3-642-18231-0.

[99]

Yu. V. Prokhorov and A. N. Shiryaev, Probability Theory III: Stochastic Calculus,, Springer, (1998). doi: 10.1007/978-3-662-03640-2.

[100]

E. D. Conway, Stochastic equations with discontinuous drift,, Trans. Am. Math. Soc., 157 (1971), 235. doi: 10.1090/S0002-9947-1971-0275532-6.

[101]

D. Stroock and S. R. S Varadhan, Diffusion processes with continuous coefficients. I,, Comm. Pure Appl. Math., 22 (1969), 345. doi: 10.1002/cpa.3160220304.

[102]

A. Ju. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations,, Math. USSR Sb., 39 (1981), 387.

[103]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift,, Probab. Theory Relat. Fields, 131 (2005), 154. doi: 10.1007/s00440-004-0361-z.

[104]

S. E. A. Mohammed, T. Nilssen and F. Proske, Sobolev differentiable stochastic flows for SDE's with singular coefficients: Applications to the transport equation,, Unpublished, (2012).

[105]

H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications,, Springer-Verlag, (1984). doi: 10.1007/978-3-642-96807-5.

[106]

Z. Schuss, Theory and Applications of Stochastic Processes,, Springer, (2010). doi: 10.1007/978-1-4419-1605-1.

[107]

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers,, International Series in Pure and Applied Mathematics. McGraw-Hill, (1978).

[108]

R. Buckdahn, Y. Ouknine and M. Quincampoix, On limiting values of stochastic differential equations with small noise intensity tending to zero,, Bull. Sci. Math., 133 (2009), 229. doi: 10.1016/j.bulsci.2008.12.005.

[109]

O. Menoukeu-Pamen, T. Meyer-Brandis and F. Proske, A Gel'fand triple approach to the small noise problem for discontinuous ODE's,, Unpublished, (2011).

[110]

G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization,, Springer, (2008).

[111]

I. Karatzas and S. E. Shreve, Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control,, Ann. Prob., 12 (1984), 819. doi: 10.1214/aop/1176993230.

[112]

I. V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures,, Theory Prob. Appl., 5 (1960), 285.

[113]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (2003). doi: 10.1007/978-3-642-14394-6.

[114]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, Springer, (1991). doi: 10.1007/978-1-4612-0949-2.

[115]

Z. Qian and W. Zheng, Sharp bounds for transition probability densities of a class of diffusions,, C.R. Acad. Sci. Paris, 335 (2002), 953. doi: 10.1016/S1631-073X(02)02579-7.

[116]

Z. Qian, F. Russo and W. Zheng, Comparison theorem and estimates for transition probability densities of diffusion processes,, Probab. Theory Relat. Fields., 127 (2003), 388. doi: 10.1007/s00440-003-0291-1.

[117]

M. Gradinaru, S. Herrmann and B. Roynette, A singular large deviations phenomenon,, Ann. Inst. Henri Poincaré, 37 (2001), 555. doi: 10.1016/S0246-0203(01)01075-5.

[118]

W. Zhang, Transition density of one-dimensional diffusion with discontinuous drift,, IEEE Trans. Automat. Contr., 35 (1990), 980. doi: 10.1109/9.58517.

show all references

References:
[1]

A. J. Van der Schaft and J. M. Schumacher, An Introduction to Hybrid Dynamical Systems,, Springer-Verlag, (2000).

[2]

R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, volume 18 of Lecture Notes in Applied and Computational Mathematics,, Springer-Verlag, (2004). doi: 10.1007/978-3-540-44398-8.

[3]

S. Banerjee and G. C. Verghese, Nonlinear Phenomena in Power Electronics,, IEEE Press, (2001). doi: 10.1109/9780470545393.

[4]

Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems,, World Scientific, (2003).

[5]

T. Puu and I. Sushko, editors, Business Cycle Dynamics: Models and Tools,, Springer-Verlag, (2006).

[6]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems. Theory and Applications,, Springer-Verlag, (2008).

[7]

N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems,, Springer, (2006).

[8]

B. Lindner, J. Garcia-Ojalvo, A. Neiman and L. Schimansky-Geier, Effects of noise in excitable systems,, Phys. Reports, 392 (2004), 321. doi: 10.1016/j.physrep.2003.10.015.

[9]

A. S. Pikovsky and J. Kurths, Coherence resonance in a noise-driven excitable system,, Phys. Rev. Lett., 78 (1997), 775. doi: 10.1103/PhysRevLett.78.775.

[10]

L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic resonance,, Rev. Modern Phys., 70 (1998), 223. doi: 10.1103/RevModPhys.70.223.

[11]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Kluwer Academic Publishers., (1988). doi: 10.1007/978-94-015-7793-9.

[12]

M. Wiercigroch and B. De Kraker, Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities,, Singapore, (2000). doi: 10.1142/9789812796301.

[13]

B. Brogliato, Nonsmooth Mechanics: Models, Dynamics and Control,, Springer-Verlag, (1999).

[14]

B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak and J. Wojewoda, Chaotic Mechanics in Systems with Impacts and Friction,, World Scientific, (1999). doi: 10.1142/9789812798565.

[15]

J. Awrejcewicz and C. Lamarque, Bifurcation and Chaos in Nonsmooth Mechanical Systems,, World Scientific, (2003). doi: 10.1142/9789812564801.

[16]

R. A. Ibrahim, Vibro-Impact Dynamics., volume 43 of Lecture Notes in Applied and Computational Mechanics,, Springer, (2009). doi: 10.1007/978-3-642-00275-5.

[17]

C. K. Tse, Complex Behavior of Switching Power Converters,, CRC Press, (2003). doi: 10.1109/JPROC.2002.1015006.

[18]

A. F. Filippov, Differential equations with discontinuous right-hand side., Mat. Sb., 51 (1960), 99.

[19]

P. Casini, O. Giannini and F. Vestroni, Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics,, Nonlinear Dyn., 46 (2006), 259. doi: 10.1007/s11071-006-9041-0.

[20]

A. C. J. Luo and B. C. Gegg, Stick and non-stick periodic motions in periodically forced oscillators with dry friction,, J. Sound Vib., 291 (2006), 132. doi: 10.1016/j.jsv.2005.06.003.

[21]

M. Johansson, Piecewise Linear Control Systems., volume 284 of Lecture Notes in Control and Information Sciencesm, Springer-Verlag, (2003). doi: 10.1007/3-540-36801-9.

[22]

M. di Bernardo, K. H. Johansson and F. Vasca, Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations,, Int J. Bifurcation Chaos, 11 (2001), 1121. doi: 10.1142/S0218127401002584.

[23]

Z. Schuss, Theory and Applications of Stochastic Differential Equations,, Wiley, (1980).

[24]

C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,, Springer-Verlag, (1985).

[25]

M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems,, Springer, (2012). doi: 10.1007/978-3-642-25847-3.

[26]

J. Grasman and O. A. van Herwaarden, Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications,, Springer, (1999). doi: 10.1007/978-3-662-03857-4.

[27]

, Special Issue on Nonsmooth Systems,, Phys. D, 241 (2012).

[28]

M. Dutta, H. E. Nusse, E. Ott, J. A. Yorke and G. Yuan, Multiple attractor bifurcations: A source of unpredictability in piecewise smooth systems,, Phys. Rev. Lett., 83 (1999), 4281. doi: 10.1103/PhysRevLett.83.4281.

[29]

T. C. L. Griffin, Dynamics of Stochastic Nonsmooth Systems,, PhD thesis, (2005).

[30]

T. Griffin and S. Hogan, Dynamics of discontinuous systems with imperfections and noise,, in IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, (2005), 275. doi: 10.1007/1-4020-3268-4_26.

[31]

P. Glendinning, The border collision normal form with stochastic switching surface,, SIAM J. Appl. Dyn. Sys., 13 (2014), 181. doi: 10.1137/130931643.

[32]

M. A. Hassouneh, E. H. Abed and H. E. Nusse, Robust dangerous border-collision bifurcations in piecewise smooth systems,, Phys. Rev. Lett., 92 (2004). doi: 10.1103/PhysRevLett.92.070201.

[33]

A. Ganguli and S. Banerjee, Dangerous bifurcation at border collision: When does it occur?,, Phys. Rev. E., 71 (2005). doi: 10.1103/PhysRevE.71.057202.

[34]

Y. Do, A mechanism for dangerous border collision bifurcations,, Chaos Solitons Fractals, 32 (2007), 352. doi: 10.1016/j.chaos.2006.07.018.

[35]

Y. Do and H. K. Baek, Dangerous border-collision bifurcations of a piecewise-smooth map,, Comm. Pure Appl. Anal., 5 (2006), 493. doi: 10.3934/cpaa.2006.5.493.

[36]

R. Wackerbauer, Noise-induced stabilization of one-dimensional discontinuous maps,, Phys. Rev. E, 58 (1998), 3036.

[37]

L. Zhang, P. Shi, C. Wang and H. Gao, Robust $H_\infty$ filtering for switched linear discrete-time systems with polytopic uncertainties,, Int. J. Adapt. Control Signal Process, 20 (2006), 291. doi: 10.1002/acs.901.

[38]

W. Zhang, J. Hu and J. Lian, Quadratic optimal control of switched linear stochastic systems,, Syst. Contr. Lett., 59 (2010), 736. doi: 10.1016/j.sysconle.2010.08.010.

[39]

P. H. E. Tiesinga, Precision and reliability of periodically and quasiperiodically driven integrate-and-fire neurons,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.041913.

[40]

A. B. Nordmark, Non-periodic motion caused by grazing incidence in impact oscillators,, J. Sound Vib., 2 (1991), 279. doi: 10.1016/0022-460X(91)90592-8.

[41]

H. Dankowicz and A. B. Nordmark, On the origin and bifurcations of stick-slip oscillations,, Phys. D, 136 (2000), 280. doi: 10.1016/S0167-2789(99)00161-X.

[42]

M. H. Fredriksson and A. B. Nordmark, On normal form calculation in impact oscillators,, Proc. R. Soc. A, 456 (2000), 315. doi: 10.1098/rspa.2000.0519.

[43]

M. di Bernardo, C. J. Budd and A. R. Champneys, Normal form maps for grazing bifurcations in $n$-dimensional piecewise-smooth dynamical systems,, Phys. D, 160 (2001), 222. doi: 10.1016/S0167-2789(01)00349-9.

[44]

D. J. W. Simpson, J. Hogan and R. Kuske, Stochastic regular grazing bifurcations,, SIAM J. Appl. Dyn. Sys., 12 (2013), 533. doi: 10.1137/120884286.

[45]

M. F. Dimentberg and D. V. Iourtchenko, Random vibrations with impacts: A review,, Nonlinear Dyn., 36 (2004), 229. doi: 10.1023/B:NODY.0000045510.93602.ca.

[46]

M. F. Dimentberg and A. I. Menyailov, Response of a single-mass vibroimpact system to white-noise random excitation,, Z. Angew. Math. Mech., 59 (1979), 709. doi: 10.1002/zamm.19790591205.

[47]

N. Sri Namachchivaya and J. H. Park, Stochastic dynamics of impact oscillators,, J. Appl. Mech. Trans. ASME, 72 (2005), 862. doi: 10.1115/1.2041660.

[48]

J. Feng, W. Xu, H. Rong and R. Wang, Stochastic responses of Duffing-Van der Pol vibro-impact system under additive and multiplicative random excitations,, Int. J. Non-Linear Mech., 44 (2009), 51. doi: 10.1016/j.ijnonlinmec.2008.08.013.

[49]

V. F. Zhuravlev, A method for analyzing vibration-impact systems by means of special functions,, Mech. Solids, 11 (1976), 23.

[50]

P. D. Christofides and N. H. El-Farra, Control of Nonlinear and Hybrid Process Systems. Designs for Uncertainty, Constraints and Time-Delays,, Springer, (2005).

[51]

J. Raouf and H. Michalska, Robust stabilization of switched linear systems with Wiener process noise,, in 49th IEEE Conference on Decision and Control, (2010), 6493. doi: 10.1109/CDC.2010.5717694.

[52]

W. Feng and J.-F. Zhang, Stability analysis and stabilization control of multi-variable switched stochastic systems,, Automatica, 42 (2006), 169. doi: 10.1016/j.automatica.2005.08.016.

[53]

D. Chatterjee and D. Liberzon, On stability of stochastic switched systems,, in Proceedings of the 43rd IEEE Conference on Decision and Control., (2004), 4125. doi: 10.1109/CDC.2004.1429398.

[54]

E. Skafidas, R. J. Evans, A. V. Savkin and I. R. Petersen, Stability results for switched controller systems,, Automatica, 35 (1999), 553. doi: 10.1016/S0005-1098(98)00167-8.

[55]

P. Mhaskar, N. H. El-Farra and P. D. Christofides, Robust hybrid predictive control of nonlinear systems,, Automatica, 41 (2005), 209. doi: 10.1016/j.automatica.2004.08.020.

[56]

B. Hu, X. Xu, P.J. Antsaklis and A. N. Michel, Robust stabilizing control laws for a class of second-order switched systems,, Syst. Control Lett., 38 (1999), 197. doi: 10.1016/S0167-6911(99)00065-1.

[57]

Z. Sun, A robust stabilizing law for switched linear systems,, Int. J. Control, 77 (2004), 389. doi: 10.1080/00207170410001667468.

[58]

D. Liberzon, Switching in Systems and Control,, Birkhauser, (2003). doi: 10.1007/978-1-4612-0017-8.

[59]

S.-C. Tan, Y.-M. Lai and C. K. Tse, Sliding Mode Control of Switching Power Converters,, CRC Press, (2012).

[60]

J.-Q. Sun, Stochastic Dynamics and Control., volume 4 of Nonlinear Science and Complexity,, Elsevier, (2006).

[61]

Y. Niu, D. W. C. Ho and J. Lam, Robust integral sliding mode control for uncertain stochastic systems with time-varying delay,, Automatica, 41 (2005), 873. doi: 10.1016/j.automatica.2004.11.035.

[62]

L. Wu, D. W. C. Ho and C. W. Li, Stabilisation and performance synthesis for switched stochastic systems,, IET Control Theory Appl., 4 (2010), 1877. doi: 10.1049/iet-cta.2009.0179.

[63]

L. Wu, D. W. C. Ho and C. W. Li, Sliding mode control of switching hybrid systems with stochastic perturbation,, Syst. Contr. Lett., 60 (2011), 531. doi: 10.1016/j.sysconle.2011.04.007.

[64]

Y. Niu, D. W. C. Ho and X. Wang, Robust $H_\infty$ control for nonlinear stochastic systems: A sliding-mode approach,, IEEE Trans. Automat. Contr., 53 (2008), 1695. doi: 10.1109/TAC.2008.929376.

[65]

D. E. Stewart, Rigid-body dynamics with friction and impact,, SIAM Rev., 42 (2000), 3. doi: 10.1137/S0036144599360110.

[66]

M. Abadie, Dynamical simulation of rigid bodies: Modelling of frictional contact,, in Impacts in Mechanical Systems: Analysis and Modelling, (2000). doi: 10.1007/3-540-45501-9_2.

[67]

P. S. Goohpattader, S. Mettu and M. K. Chaudhury, Experimental investigation of the drift and diffusion of small objects on a surface subjected to a bias and an external white noise: Roles of Coulombic friction and hysteresis,, Langmuir, 25 (2009), 9969. doi: 10.1021/la901111u.

[68]

P. S. Goohpattader and M. K. Chaudhury, Diffusive motion with nonlinear friction: Apparently Brownian,, J. Chem. Phys., 133 (2010). doi: 10.1063/1.3460530.

[69]

P.-G. de Gennes, Brownian motion with dry friction,, J. Stat. Phys., 119 (2005), 953. doi: 10.1007/s10955-005-4650-4.

[70]

H. Hayakawa, Langevin equation with Coulomb friction,, Phys. D, 205 (2005), 48. doi: 10.1016/j.physd.2004.12.011.

[71]

H. Touchette, E. Van der Straeten and W. Just, Brownian motion with dry friction: Fokker-Planck approach,, J. Phys. A: Math. Theor., 43 (2010). doi: 10.1088/1751-8113/43/44/445002.

[72]

A. Baule, E. G. D. Cohen and H. Touchette, A path integral approach to random motion with nonlinear friction,, J. Phys. A: Math. Theor., 43 (2010). doi: 10.1088/1751-8113/43/2/025003.

[73]

A. Baule, H. Touchette and E. G. D. Cohen, Stick-slip motion of solids with dry friction subject to random vibrations and an external field,, Nonlinearity, 24 (2011), 351. doi: 10.1088/0951-7715/24/2/001.

[74]

H. Touchette, T. Prellberg and W. Just, Exact power spectra of Brownian motion with solid friction,, J. Phys. A: Math. Theor., 45 (2012). doi: 10.1088/1751-8113/45/39/395002.

[75]

D. J. W. Simpson and R. Kuske, Stochastic perturbations of periodic orbits with sliding, Submitted to: J. Nonlin. Sci., (2014).

[76]

A. Colombo, M. di Bernardo, S. J. Hogan and M. R. Jeffrey, Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems,, Phys. D, 241 (2012), 1845. doi: 10.1016/j.physd.2011.09.017.

[77]

M. di Bernardo, P. Kowalczyk and A. Nordmark, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings,, Phys. D, 170 (2002), 175. doi: 10.1016/S0167-2789(02)00547-X.

[78]

A. Colombo and M. R. Jeffrey, Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows,, SIAM J. Appl. Dyn. Sys., 10 (2011), 423. doi: 10.1137/100801846.

[79]

M. R. Jeffrey and A. Colombo, The two-fold singularity of discontinuous vector fields,, SIAM J. Appl. Dyn. Sys., 8 (2009), 624. doi: 10.1137/08073113X.

[80]

M. A. Teixeira, Stability conditions for discontinuous vector fields,, J. Differential Equations, 88 (1990), 15. doi: 10.1016/0022-0396(90)90106-Y.

[81]

T.-S. Chiang and S.-J. Sheu, Large deviation of small perturbation of some unstable systems,, Stoch. Anal. Appl., 15 (1997), 31. doi: 10.1080/07362999708809462.

[82]

R. Kuske and G. Papanicolaou, The invariant density of a chaotic dynamical system with small noise,, Phys. D, 120 (1998), 255. doi: 10.1016/S0167-2789(98)00085-2.

[83]

P. Hitczenko and G. S. Medvedev, Bursting oscillations induced by small noise,, SIAM J. Appl. Math., 69 (2009), 1359. doi: 10.1137/070711803.

[84]

Ya. Z. Tsypkin, Relay Control Systems,, Cambridge University Press, (1984).

[85]

G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems,, Prentice Hall, (2002).

[86]

R. C. Dorf and R. H. Bishop, Modern Control Systems,, Prentice Hall, (2001). doi: 10.1109/TSMC.1981.4308749.

[87]

K. J. Åström and R. M. Murray, Feedback Systems. An Introduction for Scientists and Engineers,, Princeton University Press, (2008).

[88]

K. H. Johansson, A. Rantzer and K. J. Åström, Fast switches in relay feedback systems,, Automatica, 35 (1999), 539. doi: 10.1016/S0005-1098(98)00160-5.

[89]

K. H. Johansson, A. E. Barabanov and K. J. Åström, Limit cycles with chattering in relay feedback systems,, IEEE Trans. Automat. Contr., 47 (2002), 1414. doi: 10.1109/TAC.2002.802770.

[90]

Y. Zhao, J. Feng and C. K. Tse, Discrete-time modeling and stability analysis of periodic orbits with sliding for switched linear systems,, IEEE Trans. Circuits Systems I Fund. Theory Appl., 57 (2010), 2948. doi: 10.1109/TCSI.2010.2050230.

[91]

F. Dercole, R. Ferrière, A. Gragnani and S. Rinaldi, Coevolution of slow-fast populations: Evolutionary sliding, evolutionary pseudo-equilibria and complex Red Queen dynamics,, Proc. R. Soc. B, 273 (2006), 983. doi: 10.1098/rspb.2005.3398.

[92]

F. Dercole, A. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models,, Theor. Popul. Biol., 72 (2007), 197. doi: 10.1016/j.tpb.2007.06.003.

[93]

J. A. Amador, G. Olivar and F. Angulo, Smooth and Filippov models of sustainable development: Bifurcations and numerical computations,, Differ. Equ. Dyn. Syst., 21 (2013), 173. doi: 10.1007/s12591-012-0138-2.

[94]

S. Tang, J. Liang, Y. Xiao and R. A. Cheke, Sliding bifurcations of Filippov two stage pest control models with economic thresholds,, SIAM J. Appl. Math., 72 (2012), 1061. doi: 10.1137/110847020.

[95]

R. Szalai and H. M. Osinga, Invariant polygons in systems with grazing-sliding,, Chaos, 18 (2008). doi: 10.1063/1.2904774.

[96]

M. Tanelli, G. Osorio, M. di Bernardo, S. M. Savaresi and A. Astolfi, Existence, stability and robustness analysis of limit cycles in hybrid anti-lock braking systems,, Int. J. Contr., 82 (2009), 659. doi: 10.1080/00207170802203598.

[97]

M. di Bernardo, K. H. Johansson, U. Jönsson and F. Vasca, On the robustness of periodic solutions in relay feedback systems,, in 15th Triennial World Congress, (2002).

[98]

F. Flandoli, Random Perturbations of PDEs and Fluid Dynamic Models., volume 2015 of Lecture Notes in Mathematics,, Springer, (2011). doi: 10.1007/978-3-642-18231-0.

[99]

Yu. V. Prokhorov and A. N. Shiryaev, Probability Theory III: Stochastic Calculus,, Springer, (1998). doi: 10.1007/978-3-662-03640-2.

[100]

E. D. Conway, Stochastic equations with discontinuous drift,, Trans. Am. Math. Soc., 157 (1971), 235. doi: 10.1090/S0002-9947-1971-0275532-6.

[101]

D. Stroock and S. R. S Varadhan, Diffusion processes with continuous coefficients. I,, Comm. Pure Appl. Math., 22 (1969), 345. doi: 10.1002/cpa.3160220304.

[102]

A. Ju. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations,, Math. USSR Sb., 39 (1981), 387.

[103]

N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift,, Probab. Theory Relat. Fields, 131 (2005), 154. doi: 10.1007/s00440-004-0361-z.

[104]

S. E. A. Mohammed, T. Nilssen and F. Proske, Sobolev differentiable stochastic flows for SDE's with singular coefficients: Applications to the transport equation,, Unpublished, (2012).

[105]

H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications,, Springer-Verlag, (1984). doi: 10.1007/978-3-642-96807-5.

[106]

Z. Schuss, Theory and Applications of Stochastic Processes,, Springer, (2010). doi: 10.1007/978-1-4419-1605-1.

[107]

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers,, International Series in Pure and Applied Mathematics. McGraw-Hill, (1978).

[108]

R. Buckdahn, Y. Ouknine and M. Quincampoix, On limiting values of stochastic differential equations with small noise intensity tending to zero,, Bull. Sci. Math., 133 (2009), 229. doi: 10.1016/j.bulsci.2008.12.005.

[109]

O. Menoukeu-Pamen, T. Meyer-Brandis and F. Proske, A Gel'fand triple approach to the small noise problem for discontinuous ODE's,, Unpublished, (2011).

[110]

G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization,, Springer, (2008).

[111]

I. Karatzas and S. E. Shreve, Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control,, Ann. Prob., 12 (1984), 819. doi: 10.1214/aop/1176993230.

[112]

I. V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures,, Theory Prob. Appl., 5 (1960), 285.

[113]

B. Øksendal, Stochastic Differential Equations: An Introduction with Applications,, Springer, (2003). doi: 10.1007/978-3-642-14394-6.

[114]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, Springer, (1991). doi: 10.1007/978-1-4612-0949-2.

[115]

Z. Qian and W. Zheng, Sharp bounds for transition probability densities of a class of diffusions,, C.R. Acad. Sci. Paris, 335 (2002), 953. doi: 10.1016/S1631-073X(02)02579-7.

[116]

Z. Qian, F. Russo and W. Zheng, Comparison theorem and estimates for transition probability densities of diffusion processes,, Probab. Theory Relat. Fields., 127 (2003), 388. doi: 10.1007/s00440-003-0291-1.

[117]

M. Gradinaru, S. Herrmann and B. Roynette, A singular large deviations phenomenon,, Ann. Inst. Henri Poincaré, 37 (2001), 555. doi: 10.1016/S0246-0203(01)01075-5.

[118]

W. Zhang, Transition density of one-dimensional diffusion with discontinuous drift,, IEEE Trans. Automat. Contr., 35 (1990), 980. doi: 10.1109/9.58517.

[1]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757

[2]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

[3]

Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783

[4]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447

[5]

Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879

[6]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[7]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73

[8]

Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050

[9]

Sandra Ricardo, Witold Respondek. When is a control system mechanical?. Journal of Geometric Mechanics, 2010, 2 (3) : 265-302. doi: 10.3934/jgm.2010.2.265

[10]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[11]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135

[12]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-28. doi: 10.3934/dcdsb.2019103

[13]

Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471

[14]

Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1189-1206. doi: 10.3934/dcdsb.2017058

[15]

Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063

[16]

Dan Liu, Shigui Ruan, Deming Zhu. Bifurcation analysis in models of tumor and immune system interactions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 151-168. doi: 10.3934/dcdsb.2009.12.151

[17]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997

[18]

Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047

[19]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140

[20]

Tudor Bînzar, Cristian Lăzureanu. A Rikitake type system with one control. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1755-1776. doi: 10.3934/dcdsb.2013.18.1755

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]