On differentiability with respect to parameter of solutions to convex optimal control problems subject to state space constraints
The author investigates a family \(\{C_ h\}\) of optimal control problems depending on a real parameter h. The problem \(C_ h\) consists in minimizing the functional \(F(u,x,h)=\int^{T}_{0}f(u(t),x(t),h)dt\) subject to \(\overset\circ x(t) = A(h)x(t)+B(h)u(t)\), \(x(0)=x^ 0\), \(G(h)x(t)+g(h)\leq 0\) for all \(t\in [0,T]\), where \(u(t)\in R^ m\), \(x(t)\in R^ n\), \(t\geq 0\), \(f:R^ m\times R^ n\times R^ l\to R^ l\), A(h), B(h) are for each h matrices of the type \(n\times n\), \(n\times m\) respectively, \(G:R^ l\to R^{\ell +n}\), \(g:R^ l\to R^{\ell}\) and h is a real parameter, \(h\in [0,\delta]\). Under some regularity conditions posed on the data of this problem, the differentiability of its solutions with respect to h is investigated.
control problems depending on a real parameter
scientific article; zbMATH DE number 3881367