The Bogolyubov theorem with a differential inclusion as constraint
This paper deals with the problem of the calculus of variations in \(\mathbb{R}^n\), \[ I_F(x)=\min\Biggl[\int^1_0 f_F(t,x(t),\dot x(t));\;x(0)=x_0,\;x(1)= x_1\Biggr]\tag{\(*\)} \] and its convex relaxation \[ I^{**}_F(x)=\min\Biggl[\int^1_0 f^{**}_F(t,x(t),\dot x(t));\;x(0)=x_0,\;x(1)=x_1\Biggr],\tag{\(**\)} \] where \(f^{**}_F\) is a second conjugate of the function \(f_F\) with respect to its last argument. It has been shown that if there is an inclusion \(\dot x\in F(t,x(t))\), with right-hand side \(F(t,x)\in\mathbb{R}^n\) strictly convex and nonempty interior, then the problem \((**)\) serves as a convex relaxation of the problem \((*)\), where \[ f_F(t,x,u):=\begin{cases} f(t,x,u),\quad & u\in F(t,x),\\ +\infty,\quad & u\not\in F(t,x).\end{cases} \] The above results have been obtained on the base of bang-bang form of the above-mentioned differential inclusion. A very important property of the presented investigations is that each solution of the problem \((**)\) can be approximated to any accuracy in the uniform metric by close solution to the problem \((*)\).
optimal problem of multivalued differential equations
scientific article; zbMATH DE number 917101
scientific article