Existence of optimal control without convexity and a bang-bang theorem for linear volterra equations

Abstract

We consider a Mayer problem of optimal control monitored by an integral equation of Volterra type: $$x(t) = x(t_1 ) + \int_{t_1 }^t { [h(t,s)x(s) + g(t, s)f(s, u(s))] ds,} $$ where the measurable control functionu satisfies a constraint of the formu(t) ∈U(t) ⊂Em,t1≤t≤t2, andg is a continuous kernel.