Volterra funktional equations in the stability problem for the existence of global solutions of distributed controlled systems

Earlier the author proposed a rather general form of describing controlled initial– boundary value problems (CIBVPs) by means of Volterra functional equations (VFE)
$$ z(t)=f\left(t,A[z](t),v(t)\right) ,\quad t\equiv \{t^{1},\ldots,t^{n}\} \in \Pi\subset\mathbf{R}^n ,\quad z\in L_p^m \equiv \left(L_p\left( \Pi \right)\right)^m, $$
where $ f (.,.,.): \Pi \times \mathbf{R}^l \times \mathbf{ R}^s \rightarrow \mathbf{R}^m $; $ v(.) \in \mathcal{D} \subset L_k^s $ — control function; $ A:L_p^m\left( \Pi \right) \rightarrow L_q^l\left( \Pi \right) $ — linear operator; the operator $A$ is a Volterra operator for some system $T$ of subsets of the set $ \Pi $ in the following sense: for any $ H \in T ,$ the restriction $ \left. A \left[z \right] \right |_H $ does not depend on the values of $ z|_{\Pi \backslash H} $ (this definition of the Volterra operator is a direct multidimensional generalization of the well-known Tikhonov definition of a functional Volterra type operator). Various CIBVP (for nonlinear hyperbolic and parabolic equations, integro-differential equations, equations with delay, etc.) are reduced by the method of conversion the main part to such functional equations. The transition to equivalent VFE-description of CIBVP is adequate to many problems of distributed optimization. In particular, the author proposed (using such description) a scheme for obtaining sufficient stability conditions (under perturbations of control) of the existence of global solutions for CIBVP. The scheme uses continuation local solutions of functional equation (that is, solutions on the sets $H\in T$). This continuation is realized with the help of the chain $\{H_{1}\subset H_{2}\subset\ldots\subset H_{\mathbf{k}-1}\subset H_{\mathbf{k}}\equiv\Pi\},$ where $H_i\in T, i=\overline{1,\mathbf{k}}.$ A special local existence theorem is applied. This theorem is based on the principle of contraction mappings. In the case $ p = q = k = \infty $ under natural assumptions, the possibility of applying this principle is provided by the following: the right-hand side operator $ F_{v} [z \left (. \right)] \left (t \right) \equiv f \left(t, A [z] (t), v (t) \right) $ satisfies the Lipschitz condition in the operator form with the quasi-nilpotent «Lipschitz operator». This allows (using well-known results of functional analysis) to introduce in the space $ L_{\infty}^{m}(H) $ such an equivalent norm in which the operator of the right-hand side will be contractive. In the general case $ 1 \leq p, q, k \leq \infty $ (this case covers a much wider class of CIBVP), the operator $ F_{v},$ as a rule, does not satisfy such Lipschitz condition. From the results obtained by the author earlier, it follows that in this case there also exists an equivalent norm of the space $ L_p^m(H) ,$ for which the operator $ F_{v} $ is a contraction operator. The corresponding basic theorem (equivalent norm theorem) is based on the notion of equipotential quasi-nilpotency of a family of linear operators, acting in a Banach space. This article shows how this theorem can be applied to obtain sufficient stability conditions (under perturbations of control) of the existence of global solutions of VFE.