Hamilton-Jacobi equation related to a control problem (Q2711379)

Continuing previous work in [\textit{C. Davideanu}, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, N. Ser., Secţ. Ia 32, No. 3, 63-74 (1986; Zbl 0614.49020); ``Differential equations and control theory'', Pitman Res. Notes Math. Ser. 250, 63-68 (1991; Zbl 0818.49003)], the author uses the so-called ``fractional steps scheme'' for the associated Hamilton-Jacobi-Bellman equation to solve optimal control problems which consist in the minimization of functionals of the form: NEWLINE\[NEWLINE C(y(.);u(.)):=\int_0^T[g(y(t))+h(u(t))] dt+ \varphi_0(y(T)), \;(y,u)\in W^{1,2}(0,T;R^n)\times U, NEWLINE\]NEWLINE subject to: NEWLINE\[NEWLINE y'(t)+(F+\beta)(y(t))\ni Bu(t) \text{ a.e. }(0,T), \quad y(0)=x\in D(\beta),NEWLINE\]NEWLINE NEWLINE\[NEWLINE U:=\{u(.)\in W^{1,2}(0,T;R^m); \quad \|Bu\|^2_{L_2(0,T;R^n)}\leq A\}. NEWLINE\]NEWLINE The main results of the paper seem to be Theorems 3.1, 3.2 stating the convergence of the fractional steps scheme in [\textit{V. Barbu}, ``Mathematical methods in optimization of differential systems'' (1994; Zbl 0819.49002)] for the associated Hamilton-Jacobi equation: NEWLINE\[NEWLINE \varphi_t(t,x)+h^*(-B^*\varphi_x(t,x))+ \langle (F+\beta)(x),\varphi_x(t,x)\rangle= g(x), \quad \varphi(0,x)=\varphi_0(x). NEWLINE\]NEWLINE The method is applied on two examples whose value functions are obtained in explicit form.