On asymptotic exit-time control problems lacking coercivity
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 4, pp. 957-982.

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.

DOI: 10.1051/cocv/2014003
Classification: 49J15, 49N25, 93C10, 49L20, 49L25, 93D20
Keywords: exit-time problems, impulsive optimal control problems, viscosity solutions, asymptotic controllability
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Motta, M.; Sartori, C. On asymptotic exit-time control problems lacking coercivity. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 4, pp. 957-982. doi : 10.1051/cocv/2014003. http://archive.numdam.org/articles/10.1051/cocv/2014003/

[1] A. Bacciotti, Andrea and L. Rosier, Liapunov functions and stability in control theory. Second edition. Commun. Control Engrg. Ser. Springer-Verlag, Berlin (2005). | MR | Zbl

[2] M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997). | MR | Zbl

[3] 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). | MR | Zbl

[4] A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls. Boll. Un. Mat. Ital. B 7 (1988) 641-656. | MR | Zbl

[5] 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). | MR | Zbl

[6] P. Cannarsa and G. Da Prato, Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton-Jacobi equations. SIAM J. Control and Optim. 27 (1989) 861-875. | MR | Zbl

[7] P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function. J. Calc. Var. Partial Differ. Eqs. 3 (1995) 273-298. | MR | Zbl

[8] F. Da Lio, On the Bellman equation for infinite horizon problems with unbounded cost functional. J. Appl. Math. Optim. 41 (2000) 171-197. | MR | Zbl

[9] M. Garavello and P. Soravia, Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost. Nonlinear Differ. Equ. Appl. 11 (2004) 271-298. | MR | Zbl

[10] M. Malisoff, 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. | MR | Zbl

[11] M. Miller and E.Y. Rubinovich, Impulsive control in continuous and discrete-continuous systems. Kluwer Academic/Plenum Publishers, New York (2003). | MR | Zbl

[12] M. Motta, Viscosity solutions of HJB equations with unbounded data and characteristic points. Appl. Math. Optim. 4 (2004) 1-26. | MR | Zbl

[13] M. Motta and M, F. Rampazzo, State-constrained control problems with neither coercivity nor L1 bounds on the controls. Ann. Mat. Pura Appl. 4 (1999) 117-142. | MR | Zbl

[14] M. Motta and F. Rampazzo, Asymptotic controllability and optimal control. J. Differ. Eqs. 254 (2013) 2744-2763. | MR | Zbl

[15] M. Motta and C. Sartori, Exit time problems for nonlinear unbounded control systems. Discrete Contin. Dyn. Syst. 5 (1999) 137-156. | MR | Zbl

[16] M. Motta and C. Sartori, The value function of an asymptotic exit-time optimal control problem. Nonlinear Differ. Equ. Appl. Springer (2014). | MR

[17] F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bellman equations with fast gradient-dependence. Indiana Univ. Math. J. 49 (2000) 1043-1077. | MR | Zbl

[18] 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. | MR | Zbl

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