Optimization Techniques for State-Constrained Control and Obstacle Problems

The design of control laws for systems subject to complex state constraints still presents a significant challenge. This paper explores a dynamic programming approach to a specific class of such problems, that of reachability under state constraints. The problems are formulated in terms of nonstandard minmax and maxmin cost functionals, and the corresponding value functions are given in terms of Hamilton-Jacobi-Bellman (HJB) equations or variational inequalities. The solution of these relations is complicated in general; however, for linear systems, the value functions may be described also in terms of duality relations of convex analysis and minmax theory. Consequently, solution techniques specific to systems with a linear structure may be designed independently of HJB theory. These techniques are illustrated through two examples.

[1]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[2]  K Fan,et al.  Minimax Theorems. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[3]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[4]  George Leitmann.  OPTIMALITY AND REACHABILITY WITH FEEDBACK CONTROL , 1982 .

[5]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[6]  Robert J. Elliott,et al.  Viscosity solutions and optimal control , 1987 .

[7]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[8]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[9]  P. L. Lions.  Viscosity solutions and optimal control , 1992 .

[10]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[11]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[12]  Andrei I. Subbotin,et al.  Generalized solutions of first-order PDEs - the dynamical optimization perspective , 1994, Systems and control.

[13]  J. Herskowitz,et al.  Proceedings of the National Academy of Sciences, USA , 1996, Current Biology.

[14]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[15]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[16]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[17]  John Lygeros,et al.  Controllers for reachability specifications for hybrid systems , 1999, Autom..

[18]  Pravin Varaiya,et al.  Reach Set Computation Using Optimal Control , 2000 .

[19]  P. Varaiya,et al.  Dynamic Optimization for Reachability Problems , 2001 .

[20]  John Lygeros.  On the relation of reachability to minimum cost optimal control , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[21]  Pravin Varaiya,et al.  REACHABILITY UNDER STATE CONSTRAINTS - THE ELLIPSOIDAL TECHNIQUE , 2002 .

[22]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[23]  Alexander B. Kurzhanski,et al.  Control Synthesis for State Constrained Systems and Obstacle Problems , 2004 .

[24]  P. Varaiya,et al.  On Some Nonstandard Dynamic Programming Problems of Control Theory , 2005 .

[25]  Ian M. Mitchell,et al.  A Toolbox of Level Set Methods , 2005 .