A Unified Treatment of Some Nonstandard Problems in Dynamics Optimization

Abstract

We provide a theory of necessary conditions for a class of dynamic optimization problems (P) described as follows:

$$Minimize\,\sum\limits_{i = 1}^N {\int\limits_{t_i^ - }^{t_i^ + } {{l_i}} (t,{x_i}(t),\,u{}_i(t))} dt\, + \,h(\{ t_i^ - ,t_i^ + ,{x_i}(t_i^ + )\} )$$

subject to

$$\begin{array}{*{20}{c}}{{{\dot x}_i}\, = \,{\phi _i}(t,{x_i}(t),\,{u_i}(t))\,a.e\,t \in \,[t_i^ - ,t_i^ + ]} \\ {{x_i}(t)\, \in \,{X_i}(t)\,a.e\,t \in \,[t_i^ - ,t_i^ + ]} \\ {{u_i}(t)\, \in \,{\Omega _i}(t)\,a.e\,t \in \,[t_i^ - ,t_i^ + ]} \end{array}$$

for i = 1, ... , N, and

$$\{ \,t_i^ - ,t_i^ + ,{x_i}(t_i^ - ),\,{x_i}(t_i^ + )\} _{i = 1}^N\, \in \Lambda $$