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Minimum time problem with impulsive and ordinary controls

This research is partially supported by the INdAM-GNAMPA Project 2017 "Optimal impulsive control: higher order necessary conditions and gap phenomena"; and by the Padova University grant PRAT 2015 "Control of dynamics with reactive constraints"

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  • Given a nonlinear control system depending on two controls $u$ and $v$, with dynamics affine in the (unbounded) derivative of $u$ and a closed target set $\mathcal{S}$ depending both on the state and on the control $u$, we study the minimum time problem with a bound on the total variation of $u$ and $u$ constrained in a closed, convex set $U$, possibly with empty interior. We revisit several concepts of generalized control and solution considered in the literature and show that they all lead to the same minimum time function $T$. Then we obtain sufficient conditions for the existence of an optimal generalized trajectory-control pair and study the possibility of Lavrentiev-type gap between the minimum time in the spaces of regular (that is, absolutely continuous) and generalized controls. Finally, under a convexity assumption on the dynamics, we characterize $T$ as the unique lower semicontinuous solution of a regular HJ equation with degenerate state constraints.

    Mathematics Subject Classification: Primary: 49N25, 49L25; Secondary: 34H5, 49J15.


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  •   S. Aronna  and  F. Rampazzo , $L^1$ limit solutions for control systems, J. Differential Equations, 258 (2015) , 954-979.  doi: 10.1016/j.jde.2014.10.013.
      M.S. Aronna , M. Motta  and  F. Rampazzo , Infimum gaps for limit solutions, Set-Valued Var. Anal., 23 (2015) , 3-22.  doi: 10.1007/s11228-014-0296-1.
      A. Arutyunov , D. Karamzin  and  F. L. Pereira , On a generalization of the impulsive control concept: Controlling system jumps, Discrete Contin. Dyn. Syst., 29 (2011) , 403-415.  doi: 10.3934/dcds.2011.29.403.
      J. P. Aubin, and H. Frankowska, Set-valued Analysis, Reprint of the 1990 edition. Modern Birkhuser Classics. Birkhuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.
      D. Azimov  and  R. Bishop , New trends in astrodynamics and applications: Optimal trajectories for space guidance, Ann. New York Acad. Sci., 1065 (2005) , 189-209.  doi: 10.1196/annals.1370.002.
      A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, AIMS Series on Applied Mathematics, 2. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.
      A. Bressan  and  F. Rampazzo , On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. B, 2 (1988) , 641-656. 
      P. Bousquet , C. Mariconda  and  G. Treu , On the Lavrentiev phenomenon for multiple integral scalar variational problems, J. Funct. Anal., 266 (2014) , 5921-5954.  doi: 10.1016/j.jfa.2013.12.020.
      P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58. Birkhuser Boston, Inc., Boston, MA, 2004.
      A. Catllá , D. Schaeffer , T. Witelski , E. Monson  and  A. Lin , On spiking models for synaptic activity and impulsive differential equations, SIAM Rev., 50 (2008) , 553-569.  doi: 10.1137/060667980.
      F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics vol. 178, Springer-Verlag, New York, 1998.
      V. A. Dykhta, Impulse-trajectory extension of degenerated optimal control problems. The Lyapunov functions method and applications, IMACS Ann. Comput. Appl. Math., 8, Baltzer, Basel, (1990), 103-109.
      H. Frankowska  and  S. Plaskacz , Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints, J. Math. Anal. Appl., 251 (2000) , 818-838.  doi: 10.1006/jmaa.2000.7070.
      P. Gajardo , C.H. Ramirez  and  A. Rapaport , Minimal time sequential batch reactors with bounded and impulse controls for one or more species, SIAM J. Control Optim., 47 (2008) , 2827-2856.  doi: 10.1137/070695204.
      M. Guerra  and  A. Sarychev , Fréchet generalized trajectories and minimizers for variational problems of low coercivity, J. Dyn. Control Syst., 21 (2015) , 351-377.  doi: 10.1007/s10883-014-9231-x.
      V. I. Gurman , Optimal processes of singular control, Automat. Remote Control, 26 (1965) , 783-792. 
      D. Y. Karamzin , V. A. de Oliveira , F. L. Pereira  and  G. N. Silva , On the properness of an impulsive control extension of dynamic optimization problems, ESAIM Control Optim. Calc. Var., 21 (2015) , 857-875.  doi: 10.1051/cocv/2014053.
      V. F. Krotov, Global Methods in Optimal Control Theory, Monographs and Textbooks in Pure and Applied Mathematics, 195. Marcel Dekker, Inc., New York, 1996.
      K. Kunisch  and  Z. Rao , Minimal time problem with impulsive controls, Appl. Math. Optim., 75 (2017) , 75-97.  doi: 10.1007/s00245-015-9324-2.
      T. T. Le Thuy  and  A. Marigonda , Small-time local attainability for a class of control systems with state constraints, ESAIM Control Optim. Calc. Var., 23 (2017) , 1003-1021.  doi: 10.1051/cocv/2016022.
      B. M. Miller, Optimization of dynamical systems with generalized control. (Russian) Avtomat. i Telemekh., 1989, 23-34; translation in Automat. Remote Control, 50 (1989), 733-742. doi: MR1016198.
      B. M. Miller, The method of discontinuous time substitution in problems of the optimal control of impulse and discrete-continuous systems. (Russian) Avtomat. i Telemekh., 1993, 3-32; translation in Automat. Remote Control, 54 (1993), 1727-1750.
      B. M. Miller , The generalized solutions of nonlinear optimization problems with impulse control, SIAM J. Control Optim., 34 (1996) , 1420-1440.  doi: 10.1137/S0363012994263214.
      B. M. Miller and E. Ya. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic/Plenum Publishers, New York, 2003. doi: 10.1007/978-1-4615-0095-7.
      B. M. Miller and E. Ya. Rubinovich, Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations, Translation of Avtomat. i Telemekh., 2013, 56-103. Autom. Remote Control, 74 (2013), 1969-2006. doi: 10.1134/S0005117913120047.
      M. Motta  and  F. Rampazzo , Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995) , 269-288. 
      M. Motta  and  F. Rampazzo , Dynamic programming for nonlinear systems driven by ordinary and impulsive controls, SIAM J. Control Optim., 34 (1996) , 199-225.  doi: 10.1137/S036301299325493X.
      M. Motta  and  C. Sartori , Minimum time with bounded energy, minimum energy with bounded time, SIAM J. Control Optim., 42 (2003) , 789-809.  doi: 10.1137/S0363012902385284.
      M. Motta  and  C. Sartori , Semicontinuous viscosity solutions to mixed boundary value problems with degenerate convex Hamiltonians, Nonlinear Anal., 49 (2002) , Ser. A: Theory Methods, 905-927.  doi: 10.1016/S0362-546X(01)00137-7.
      M. Motta  and  C. Sartori , On asymptotic exit-time control problems lacking coercivity, ESAIM Control Optim. Calc. Var., 20 (2014) , 957-982.  doi: 10.1051/cocv/2014003.
      M. Motta  and  C. Sartori , On $L^1$ limit solutions in impulsive control, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018) , 1201-1218.  doi: 10.3934/dcdss.2018068.
      M. Motta  and  C. Sartori , Lack of BV bounds for impulsive control systems, J. Math. Anal. Appl., 461 (2018) , 422-450.  doi: 10.1016/j.jmaa.2018.01.019.
      F. Rampazzo  and  C. Sartori , The minimum time function with unbounded controls, J. Math. Systems Estim. Control, 8 (1998) , 1-34. 
      A. Razmadzé , Sur les solutions discontinues dans le calcul des variations, (French) Math. Ann., 94 (1925) , 1-52.  doi: 10.1007/BF01208643.
      R. Rishel , An extended Pontryagin principle for control systems whose control laws contain measures, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965) , 191-205.  doi: 10.1137/0303016.
      R. T. Rockafellar, Convex Analysis, Princeton, NJ: Princeton University Press, Princeton, NJ, 1997.
      P. Soravia , Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. Ⅱ. Equations of control problems with state constraints, Differential Integral Equations, 12 (1999) , 275-293. 
      J. Warga , Relaxed variational problems, J. Math. Anal. Appl., 4 (1962) , 111-128.  doi: 10.1016/0022-247X(62)90033-1.
      J. Warga , Variational problems with unbounded controls, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1966) , 424-438.  doi: 10.1137/0303028.
      A. J. Zaslavski , Nonoccurrence of the Lavrentiev phenomenon for many optimal control problems, SIAM J. Control Optim., 45 (2006) , 1116-1146.  doi: 10.1137/050640370.
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