Optimal Control Systems with Stochastic Boundary Conditions and State Equations

We consider here control problems of optimization with stochastic boundary conditions that are obtained by allowing; boundary data and certain constraints to depend upon a random variable π = π( a ), α ϵ I 1 π continuous over I 1 . We also allow the state equations to depend upon a random variable or a stochastic process η = η( t , b ), b ϵ I 2 , with suitably smooth sample paths. We take for admissible controls measurable functions u = u ( t ) of t only with values in a compact control set U ( t ) ⊂ E m . Hence the admissible trajectories x = x ( t , a , b ) are stochastic processes, that is, for each t , x is a random variable over I = I 1 × I 2 with respect to a given probability measure P over I . We take as our cost functional Ek [ t 2 , x ( t 2 , a , b )] where E denotes expectation with respect to P . We state an existence theorem for such systems, and give necessary conditions that an optimal pair x 0 , u 0 must satisfy. We also discuss applications of these results to various models in operations research.