Optimal Control of Differential-Algebraic Inclusions

Abstract

In this paper we consider the following dynamic optimization problem ( P ) governed by differential-algebraic inclusions:

minimize $J [ x, z ] := \varphi ( x ( a ) , x( b )) + \int^ b_ a f ( x ( t ) , x( t - \Delta), \dot z ( t ) , t) dt$     (1)

subject to the constraints

$\dot z ( t ) \in F ( x ( t ) , x( t - \Delta), t),~~$ a.e. $t \in [ a, b ]$,     (2)

$z ( t ) = x ( t ) + Ax( t - \Delta),~\qquad t\in [ a, b ]$,     (3)

$x ( t ) = c ( t ),\qquad t$$\in$ [a - Δ,a,)     (4)

$( x ( a ) , x( b )) \in 4 \Omega \subset {\rm I\!R}^{2 n}$,     (5)

where $x : [a - \Delta , b] \to {\rm I\!R}^n$ is continuous on ($a - \Delta , a$) and [ a, b ] (with a possible jump at t = a), and where z ( t ) is absolutely continuous on [ a, b ]. We always assume that $F : {\rm I\!R}^ n \times {\rm I\!R}^n \times [ a, b ] \rightrightarrows {\rm I\!R}^n$ is a set-valued mapping of closed graph, that 4 is a closed set, that $\Delta > 0$ is a constant delay , and that A is a constant n × n matrix.