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 I₁π continuous over I₁. We also allow the state equations to depend upon a random variable or a stochastic process η = η(t, b), b ϵ I₂, 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 = I₁ × I₂ with respect to a given probability measure P over I. We take as our cost functional Ek[t₂, x(t₂, 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 x₀, u₀ must satisfy. We also discuss applications of these results to various models in operations research.