Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T13:19:24.678Z Has data issue: false hasContentIssue false
  • English
  • Français

On asymptotic exit-time control problems lacking coercivity∗

Published online by Cambridge University Press:  05 August 2014

M. Motta  and
C. Sartori
M. Motta
Affiliation:
Dipartimento di Matematica, Via Trieste, 63 – 35121 Padova, Italy. [email protected]; [email protected]
C. Sartori
Affiliation:
Dipartimento di Matematica, Via Trieste, 63 – 35121 Padova, Italy. [email protected]; [email protected]
Get access

Abstract

The research on a class of asymptotic exit-time problems with a vanishing Lagrangian, begun in [M. Motta and C. Sartori, Nonlinear Differ. Equ. Appl. Springer (2014).] for the compact control case, is extended here to the case of unbounded controls and data, including both coercive and non-coercive problems. We give sufficient conditions to have a well-posed notion of generalized control problem and obtain regularity, characterization and approximation results for the value function of the problem.

Keywords

Exit-time problemsimpulsive optimal control problemsviscosity solutionsasymptotic controllability
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 4 , October 2014 , pp. 957 - 982
Copyright
© EDP Sciences, SMAI, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A. Bacciotti, Andrea and L. Rosier, Liapunov functions and stability in control theory. Second edition. Commun. Control Engrg. Ser. Springer-Verlag, Berlin (2005). Google Scholar
M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997). Google Scholar
A. Bressan and B. Piccoli, Introduction to the mathematical theory of control. Vol. 2. AIMS Ser. Appl. Math. Amer. Institute of Math. Sci. AIMS, Springfield, MO (2007). Google Scholar
Bressan, A. and Rampazzo, F., On differential systems with vector-valued impulsive controls. Boll. Un. Mat. Ital. B 7 (1988) 641–656. Google Scholar
D.A. Carlson and A. Haurie, Infinite horizon optimal control. Theory and applications. Vol. 290 of Lect. Notes Econom. Math. Systems. Springer-Verlag, Berlin (1987). Google Scholar
Cannarsa, P. and Prato, G. Da, Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton–Jacobi equations. SIAM J. Control and Optim. 27 (1989) 861–875. Google Scholar
Cannarsa, P. and Sinestrari, C., Convexity properties of the minimum time function. J. Calc. Var. Partial Differ. Eqs. 3 (1995) 273–298. Google Scholar
Da Lio, F., On the Bellman equation for infinite horizon problems with unbounded cost functional. J. Appl. Math. Optim. 41 (2000) 171–197. Google Scholar
Garavello, M. and Soravia, P., Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. Nonlinear Differ. Equ. Appl. 11 (2004) 271–298. Google Scholar
Malisoff, M., Bounded-from-below solutions of the Hamilton–Jacobi equation for optimal control problems with exit times: vanishing Lagrangians, eikonal equations, and shape-from-shading. Nonlinear Differ. Equ. Appl. 11 (2004) 95–122. Google Scholar
M. Miller and E.Y. Rubinovich, Impulsive control in continuous and discrete-continuous systems. Kluwer Academic/Plenum Publishers, New York (2003). Google Scholar
Motta, M., Viscosity solutions of HJB equations with unbounded data and characteristic points. Appl. Math. Optim. 4 (2004) 1–26. Google Scholar
Motta, M. and Rampazzo, M, F., State-constrained control problems with neither coercivity nor L 1 bounds on the controls. Ann. Mat. Pura Appl. 4 (1999) 117–142. Google Scholar
Motta, M. and Rampazzo, F., Asymptotic controllability and optimal control. J. Differ. Eqs. 254 (2013) 2744–2763. Google Scholar
Motta, M. and Sartori, C., Exit time problems for nonlinear unbounded control systems. Discrete Contin. Dyn. Syst. 5 (1999) 137–156. Google Scholar
M. Motta and C. Sartori, The value function of an asymptotic exit-time optimal control problem. Nonlinear Differ. Equ. Appl. Springer (2014). Google Scholar
Rampazzo, F. and Sartori, C., Hamilton-Jacobi-Bellman equations with fast gradient-dependence. Indiana Univ. Math. J. 49 (2000) 1043–1077. Google Scholar
P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton–Jacobi equations I: Equations of unbounded and degenerate control problems without uniqueness. Adv. Differ. Eqs. (1999) 275–296. Google Scholar