The author considers the boundary-value problem: \[ V_{t}-AV=\lambda \int_{0}^{T}K(t,\tau )V(\tau ,x)d\tau +f(t,x,u(t,x)), x\in Q\subset \mathbb{R}^{n}, 0<t\leq T, \] where \(A\) is an elliptic operator defined by: \[ AV(t,x)=\sum_{i,j=1}^{n}\frac{\partial }{\partial x_{i}}(a_{i,j}(x)\frac{ \partial V}{\partial x_{j}})-c(x)V(t,x), \] the coefficients satisfying the usual symmetry and coercivity conditions, \(Q\) is a domain in \(\mathbb{R}^{n}\) bounded by a piecewise smooth surface \(\gamma \), \(K(t,\tau )\) is a function which satisfies the condition \(\int_{0}^{T}\int_{0}^{T}K^{2}(t,\tau )d\tau dt=K_{0}<\infty \).
The initial condition \(V(0,\cdot )=\psi (\cdot )\in H^{1}(Q)\), and the boundary condition \[ \sum_{i,j=1}^{n}a_{i,j}(x)\frac{ \partial V}{\partial x_{i}}cos(\nu ,x_{i})+a(x)V=0, x\in \gamma, 0<t<T, \] are imposed, \(\nu \) being the outer normal. The source function \( f(t,x,u(t,x))\) belongs to \(H(Q\times (0,T))\), nonlinearly depends on the control \(u(t,x)\in H(Q\times (0,T))\), and satisfies the monotonicity condition: \(f_{u}(t,x,u(t,x)\neq 0, \forall (t,x)\in Q\times (0,T)\). The author defines a generalized (weak) solution to this boundary-value problem as an element \(V\) which belongs to \(H(Q\times (0,T))\), which satisfies a variational formulation and which satisfies the initial condition in the weak sense.
The author looks for a solution to the above problem in the form \(V(t,x)=\sum_{n=1}^{\infty }V_{n}(t)z_{n}(x)\), where \(V_{n}(t)= \int_{Q}V(t,x)z_{n}(x)dx\) are the Fourier coefficients, \(z_{n}(x)\) being the generalized eigenfunctions of the boundary-value problem: \[ \begin{aligned} &\int_{Q}(\sum_{i,j=1}^{n}a_{i,j}\frac{\partial V(t,x)}{\partial x_{j}}\frac{ \partial z_{n}(t,x)}{\partial x_{i}}+c(x)V(t,x)z_{n}(x))dx+\int_{\gamma }a(x)V(t,x)z_{n}(x)dx \\ &=\lambda _{n}\int_{Q}V(t,x)z_{n}(x)dx. \end{aligned} \] The author computes these Fourier coefficients and proves that the resulting function \( V(t,x)\) belongs to \(H(Q\times (0,T))\). He then introduces the quadratic integral functional \(J[u(t,x)]=\int_{0}^{T}\int_{Q}V(t,x)-\xi (t,x)^{2}dxdt+\beta \int_{0}^{T}\int_{Q}p^{2}[t,x,u(t,x)]dxdt\), where \(\beta >0\), \(\xi (x,t),p[t,x,u(t,x)]\in H(Q\times (0,T))\) are given functions and \( p[t,x,u(t,x)]\) nonlinearly depends on the functional variable \(u(t,x)\in H(Q\times (0,T))\) on the set of solutions to the the boundary-value problem. He looks for a control \(u^{0}(t,x)\) in the form \(u^{0}(t,x)=u[t,x,V^{0}(t,x)] \). He introduces the Bellman functional for \(J\): \(S(t,V(t,x))=min_{u(\tau )\in U,t\leq \tau \leq T,x\in Q}\{\int_{0}^{T}\int_{Q}V(t,x)-\xi (t,x)^{2}dxdt+\beta \int_{t}^{T}\int_{Q}p^{2}[\tau ,x,u(\tau ,x)]dxdt\}\), where \(V(t,x)\) is the state function, \(\xi (x,t)\) a given function, and \(U\) the set of admissible values of the control \(u(t,x)\in H(Q\times (0,T))\). To solve this problem, he applies the Bellman optimality principle. He computes the Bellman functional and its derivative with respect to time. He ends with a Cauchy problem and he analyzes the conditions which ensure its solvability.