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May  2021, 41(5): 2447-2474. doi: 10.3934/dcds.2020371

Stabilizability in optimization problems with unbounded data

Anna Chiara Lai 1, and  Monica Motta 2,,

1. 

Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Scarpa, 16, Roma 00181, Italy
2. 

Dipartimento di Matematica "Tullio Levi-Civita", Università di Padova, Via Trieste, 63, Padova 35121, Italy

* Corresponding author

Received  April 2020 Revised  September 2020 Published  May 2021 Early access  November 2020

Fund Project: This research is partially supported by the Padua University grant SID 2018 "Controllability, stabilizability and infimun gaps for control systems", prot. BIRD 187147, and by the INdAM-GNAMPA Project 2020 "Extended control problems: gap, higher order conditions and Lyapunov functions"

In this paper we extend the notions of sample and Euler stabilizability to a set of a control system to a wide class of systems with unbounded controls, which includes nonlinear control-polynomial systems. In particular, we allow discontinuous stabilizing feedbacks, which are unbounded approaching the target. As a consequence, sampling trajectories may present a chattering behaviour and Euler solutions have in general an impulsive character. We also associate to the control system a cost and provide sufficient conditions, based on the existence of a special Lyapunov function, which allow for the existence of a stabilizing feedback that keeps the cost of all sampling and Euler solutions starting from the same point below the same value, in a uniform way.

Keywords: Asymptotic stabilizability, discontinuous feedback law, optimal control, exit-time problems, impulsive control systems.
Mathematics Subject Classification: Primary: 93D05, 93D20; Secondary: 49J15, 49N25.
Citation: Anna Chiara Lai, Monica Motta. Stabilizability in optimization problems with unbounded data. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2447-2474. doi: 10.3934/dcds.2020371
References:
[1]

Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal., 7 (1983), 1163-1173.  doi: 10.1016/0362-546X(83)90049-4.

[2]

A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141.  doi: 10.1007/s00205-009-0237-6.

[3]

R. W. Brockett, Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, 27 (1983), 181-191. 

[4]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston, Inc., Boston, MA, 2004.

[5]

L. Cesari, Optimization Theory and Applications, Problems with ordinary differential equations. Applications of Mathematics, (New York), 17. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.

[6]

F. H. Clarke, Yu. S. Ledyaev, L. Rifford and R. J. Stern, Feedback stabilization and Lyapunov functions, SIAM J. Control Optim., 39 (2000), 25-48.  doi: 10.1137/S0363012999352297.

[7]

F. H. Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control, 42 (1997), 1394-1407.  doi: 10.1109/9.633828.

[8]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998.

[9]

J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization, J. Math. Systems Estim. Control, 4 (1994), 67-84. 

[10]

S. N. Dashkovskiĭ, D. V. Efimov and È. D. Sontag, Input-to-state stability and related properties of systems, Autom. Remote Control, 72 (2011), 1579-1614.  doi: 10.1134/S0005117911080017.

[11]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York, 1969.

[12]

D. Yu. Karamzin, V. A. de Oliveira, F. L. Pereira and G. N. Silva, On some extension of optimal control theory, Eur. J. Control, 20 (2014), 284-291.  doi: 10.1016/j.ejcon.2014.09.003.

[13]

C. M. Kellett and A. R. Teel, Uniform asymptotic controllability to a set implies locally Lipschitz control-Lyapunov function, Proceedings of the 39th IEEE Conference on Decision and Control, 4 (2000), 3994-3999.  doi: 10.1109/CDC.2000.912339.

[14]

N. N. Krasovskiĭ and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York, 1988.

[15]

A. C. Lai and M. Motta, Stabilizability in optimal control, NoDEA Nonlinear Differential Equations Appl., 27 (2020), Paper No. 41, 32 pp. doi: 10.1007/s00030-020-00647-7.

[16]

A. C. Lai and M. Motta, Stabilizability in impulsive optimization problems, Proceedings of the 11th IFAC Symposium on Nonlinear Control Systems, NOLCOS, Vienna, IFAC-PapersOnLine, 52 (2019), 352–357.

[17]

A. C. Lai, M. Motta and F. Rampazzo, Minimum restraint functions for unbounded dynamics: General and control-polynomial systems, Pure and Applied Functional Analysis, 1 (2016), 583-612. 

[18]

M. Malisoff, L. Rifford and E. Sontag, Global asymptotic controllability implies input-to-state stabilization, SIAM J. Control Optim., 42 (2004), 2221-2238.  doi: 10.1137/S0363012903422333.

[19]

M. Motta and F. Rampazzo, Asymptotic controllability and optimal control, Journal of Differential Equations, 254 (2013), 2744-2763.  doi: 10.1016/j.jde.2013.01.006.

[20]

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.

[21]

M. Motta and C. Sartori, Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems, Discrete Contin. Dyn. Syst., 21 (2008), 513-535.  doi: 10.3934/dcds.2008.21.513.

[22]

F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bellman equations with fast gradient-dependence, Indiana Univ. Math. J., 49 (2000), 1043-1077.  doi: 10.1512/iumj.2000.49.1736.

[23]

L. Rifford, Existence of Lipschitz and semiconcave control-Lyapunov functions, SIAM J. Control Optim., 39 (2000), 1043-1064.  doi: 10.1137/S0363012999356039.

[24]

L. Rifford, Semiconcave control-Lyapunov functions and stabilizing feedbacks, SIAM J. Control Optim., 41 (2002), 659-681.  doi: 10.1137/S0363012900375342.

[25]

E. P. Ryan, On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback, SIAM J. Control Optim., 32 (1994), 1597-1604.  doi: 10.1137/S0363012992235432.

[26]

E. D. Sontag, A Lyapunov-like characterization of asymptotic controllability, SIAM J. Control Optim., 21 (1983), 462-471.  doi: 10.1137/0321028.

[27]

E. Sontag and H. J. Sussmann, Nonsmooth control-Lyapunov functions, Proceedings of the IEEE Conference on Decision and Control, 3 (1995), 2799-2805.  doi: 10.1109/CDC.1995.478542.

[28]

E. Sontag and Y. Wang, Various results concerning set input-to-state stability, Proc. 34th IEEE Conf. Decision and Control, New Orleans, December, (1995), 1330–1335.

[29]

R. Vinter, Optimal Control, Birkhäuser, Boston, 2000.

show all references

References:
[1]

Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal., 7 (1983), 1163-1173.  doi: 10.1016/0362-546X(83)90049-4.

[2]

A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141.  doi: 10.1007/s00205-009-0237-6.

[3]

R. W. Brockett, Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, 27 (1983), 181-191. 

[4]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston, Inc., Boston, MA, 2004.

[5]

L. Cesari, Optimization Theory and Applications, Problems with ordinary differential equations. Applications of Mathematics, (New York), 17. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.

[6]

F. H. Clarke, Yu. S. Ledyaev, L. Rifford and R. J. Stern, Feedback stabilization and Lyapunov functions, SIAM J. Control Optim., 39 (2000), 25-48.  doi: 10.1137/S0363012999352297.

[7]

F. H. Clarke, Y. S. Ledyaev, E. D. Sontag and A. I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control, 42 (1997), 1394-1407.  doi: 10.1109/9.633828.

[8]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178. Springer-Verlag, New York, 1998.

[9]

J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization, J. Math. Systems Estim. Control, 4 (1994), 67-84. 

[10]

S. N. Dashkovskiĭ, D. V. Efimov and È. D. Sontag, Input-to-state stability and related properties of systems, Autom. Remote Control, 72 (2011), 1579-1614.  doi: 10.1134/S0005117911080017.

[11]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York, 1969.

[12]

D. Yu. Karamzin, V. A. de Oliveira, F. L. Pereira and G. N. Silva, On some extension of optimal control theory, Eur. J. Control, 20 (2014), 284-291.  doi: 10.1016/j.ejcon.2014.09.003.

[13]

C. M. Kellett and A. R. Teel, Uniform asymptotic controllability to a set implies locally Lipschitz control-Lyapunov function, Proceedings of the 39th IEEE Conference on Decision and Control, 4 (2000), 3994-3999.  doi: 10.1109/CDC.2000.912339.

[14]

N. N. Krasovskiĭ and A. I. Subbotin, Game-Theoretical Control Problems, Springer-Verlag, New York, 1988.

[15]

A. C. Lai and M. Motta, Stabilizability in optimal control, NoDEA Nonlinear Differential Equations Appl., 27 (2020), Paper No. 41, 32 pp. doi: 10.1007/s00030-020-00647-7.

[16]

A. C. Lai and M. Motta, Stabilizability in impulsive optimization problems, Proceedings of the 11th IFAC Symposium on Nonlinear Control Systems, NOLCOS, Vienna, IFAC-PapersOnLine, 52 (2019), 352–357.

[17]

A. C. Lai, M. Motta and F. Rampazzo, Minimum restraint functions for unbounded dynamics: General and control-polynomial systems, Pure and Applied Functional Analysis, 1 (2016), 583-612. 

[18]

M. Malisoff, L. Rifford and E. Sontag, Global asymptotic controllability implies input-to-state stabilization, SIAM J. Control Optim., 42 (2004), 2221-2238.  doi: 10.1137/S0363012903422333.

[19]

M. Motta and F. Rampazzo, Asymptotic controllability and optimal control, Journal of Differential Equations, 254 (2013), 2744-2763.  doi: 10.1016/j.jde.2013.01.006.

[20]

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.

[21]

M. Motta and C. Sartori, Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems, Discrete Contin. Dyn. Syst., 21 (2008), 513-535.  doi: 10.3934/dcds.2008.21.513.

[22]

F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bellman equations with fast gradient-dependence, Indiana Univ. Math. J., 49 (2000), 1043-1077.  doi: 10.1512/iumj.2000.49.1736.

[23]

L. Rifford, Existence of Lipschitz and semiconcave control-Lyapunov functions, SIAM J. Control Optim., 39 (2000), 1043-1064.  doi: 10.1137/S0363012999356039.

[24]

L. Rifford, Semiconcave control-Lyapunov functions and stabilizing feedbacks, SIAM J. Control Optim., 41 (2002), 659-681.  doi: 10.1137/S0363012900375342.

[25]

E. P. Ryan, On Brockett's condition for smooth stabilizability and its necessity in a context of nonsmooth feedback, SIAM J. Control Optim., 32 (1994), 1597-1604.  doi: 10.1137/S0363012992235432.

[26]

E. D. Sontag, A Lyapunov-like characterization of asymptotic controllability, SIAM J. Control Optim., 21 (1983), 462-471.  doi: 10.1137/0321028.

[27]

E. Sontag and H. J. Sussmann, Nonsmooth control-Lyapunov functions, Proceedings of the IEEE Conference on Decision and Control, 3 (1995), 2799-2805.  doi: 10.1109/CDC.1995.478542.

[28]

E. Sontag and Y. Wang, Various results concerning set input-to-state stability, Proc. 34th IEEE Conf. Decision and Control, New Orleans, December, (1995), 1330–1335.

[29]

R. Vinter, Optimal Control, Birkhäuser, Boston, 2000.

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