GENERALIZED SOLUTION OF THE DEPENDENT IMPULSIVE CONTROL SYSTEM CORRESPONDING TO VECTOR-VALUED CONTROLS OF BOUNDED VARIATION

  • Shin, Chang-Eon (Department of Mathematics, Sogang University) ;
  • Ryu, Ji-Hyun (Department of Mathematics, Sogang University)
  • Published : 2000.05.01

Abstract

This paper is concerned with the impulsive Cauchy problem where the control function u is a possibly discontinuous vector-valued function with finite total variation. We assume that the vector fields f, $g_i$(i=1,…, m) are dependent on the time variable. The impulsive Cauchy problem is of the form x(t)=f(t,x) +$\SUMg_i(t,x)u_i(t)$, $t\in$[0,T], x(0)=$\in\; R^n$, where the vector fields f, $g_i$ : $\mathbb{R}\; \times\; \mathbb{R}\; \longrightarrow\; \mathbb(R)^n$ are measurable in t and Lipschitz continuous in x, If $g_i's$ satisfy a condition that $\SUM{\mid}g_i(t_2,x){\mid}{\leq}{\phi}$ $\forallt_1\; <\; t-2,x\; {\epsilon}\;\mathbb{R}^n$ for some increasing function $\phi$, then the imput-output function can be continuously extended to measurable functions of bounded variation.

Keywords

impulsive control system;generalized solution

References

  1. Rend. Sem. mat. Univ. Padova. Generalized Solutions of Time dependent Impulsive Control Systems C.E. Shin;J. H. Ryu
  2. Journal of dynamical and contorl systems v.3 no.2 On Mappings of Bounded Variation V.V. Chistyakov
  3. Annals of Probability v.6 On the Gap between Deterministic and Stochastic Ordinary Differential Equations H.J. Sussmann
  4. J. Optim. Theory Apply. v.71 Impulsive Control Systems with Commutative Vector Fields
  5. SIAM Journal on Control, Series A v.3 Optimal Control Theory for Nonlinear Differential Equations Containing Measures W.W. Schmaedeke
  6. Lecture Notes on the Mathematical Theory A. Bressan
  7. Differential and Integral Equations v.4 On Systems of Ordinary Differential Equations with measures as Controls G. Dal Maso;F. Rampazzo
  8. Journal of KMS v.34 Generalized Solutions of Impulsive Control Systems Corresponding to Controls of Bounded Variation C.E. Shin
  9. Shocks and Dry Friction Differential Inclusions in Nonsmooth Mechanical problems M.D.P. Monteiro Marques
  10. Nonsmooth analysis and geometric methods in deterministic optimal control Impulsive Control Systems
  11. Boll. Un. Mat. Ital. Series B v.3 On Differential Systems with Vector-Valued Impulsive Controls A. Bressan;F. Rampazzo
  12. Rend. Sem. Mat. Univ. Padova v.78 On Differential Systems with Impulsive Controls
  13. Journal of optimization theory and applications v.81 no.3 Impulsive Control Systems without Commutative Assumptions
  14. Optimal Impulsive Controls with a Constraint on Total Variation F. Rampazzo