Second-order optimality conditions and regularity of Lagrange multipliers for mixed optimal control problems
One considers the following optimal control problem: find a couple of control functions $(u ,v)\in L^{\infty}(\Omega)\times L^{\infty}(\Gamma)$ such that if $y:= y(x;u,v)$ is the solution for the boundary value problem to a semieliptic partial differential equation, \[ Ay=f_1(x,y,u),\text{ in }\Omega,\partial _{\nu _A}y =f_2(x,y,v),\text{ on }\Gamma, \] the following functional gets minimized, \[ F(y;u,v)=\int _{\Omega}L(.,y(.),u(.))dx+\int _ {\Gamma}l(.,y(.),v(.))ds, \] $\min_{\substack {u,v}}F(y;u,v)$. The aim is to look for some second-order optimality conditions. In the second section of the article, there is a review of known results related to second-order necessary optimality conditions and regularity of Lagrange multipliers. One introduces the notion of a strongly optimal solution to the stated problem and under some special assumptions, one proves a result on second-order sufficient optimal conditions for the optimal control problem.
scientific article; zbMATH DE number 7382078
scientific article