Nonlinear Impulse Target Problems under State Constraint: A Numerical Analysis Based on Viability Theory

We study the problem of reaching a target without leaving a prescribed set for controlled impulse dynamics. First, we provide a numerical procedure for the approximation of the set of initial conditions from which the objective can be met. Then we show that the minimum time function associated with this target problem can be approached by a sequence of value functions for suitable discrete-time systems. This can be deduced from the fact that the epigraph of the minimum time function is the set of initial conditions from which a target can be reached without leaving a constraint set for an auxiliary impulse system. In this case, the numerical procedure for the qualitative target problem can be simplified. We provide results for estimating the convergence rate of the simplified scheme.

[1]  E. Crück,et al.  Target problems under state constraints for nonlinear controlled impulsive systems , 2002 .

[2]  P. Saint-Pierre,et al.  Numerical Schemes for Discontinuous Value Functions of Optimal Control , 2000 .

[3]  P. Saint-Pierre.  Approximation of the viability kernel , 1994 .

[4]  K. Teo,et al.  On a Class of Optimal Control Problems with State Jumps , 1998 .

[5]  Augusto Visintin,et al.  Strong convergence results related to strict convexity , 1984 .

[6]  Mark R. Greenstreet,et al.  Hybrid Systems: Computation and Control , 2002, Lecture Notes in Computer Science.

[7]  Jean-Pierre Aubin,et al.  Impulse differential inclusions: a viability approach to hybrid systems , 2002, IEEE Trans. Autom. Control..

[8]  Paolo Nistri,et al.  On open-loop and feedback attainability of a closed set for nonlinear control systems , 2002 .

[9]  P. Saint-Pierre,et al.  Optimal times for constrained nonlinear control problems without local controllability , 1997 .

[10]  Mabel Tidball,et al.  Rate of Convergence of a Numerical Procedure for Impulsive Control Problems , 1996, WNAA.

[11]  Patrick Saint-Pierre.  Hybrid Kernels and Capture Basins for Impulse Constrained Systems , 2002, HSCC.

[12]  I. Dolcetta.  On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming , 1983 .

[13]  Eva Crück,et al.  Problèmes de cible sous contraintes d'état pour des systèmes non linéaires avec sauts d'état , 2001 .

[14]  P. Cardaliaguet.  A differential game with two players and one target , 1994 .

[15]  A. Bensoussan,et al.  Hybrid control and dynamic programming , 1997 .

[16]  P. Saint-Pierre,et al.  Set-Valued Numerical Analysis for Optimal Control and Differential Games , 1999 .

[17]  G. Barles,et al.  Deterministic Impulse Control Problems , 1985 .