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

V. I. Sumin

Vestnik rossijskih universitetov. Matematika, Tome 25 (2020) no. 132, pp. 422-440

Résumé

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.

Détail

Keywords: functional Volterra equation, controlled distributed system, stability of the existence of global solutions, equipotential quasinilpotent family of the operators, equivalent norm theorem.

Bibliographie

[1] V. I. Sumin, “Funktsionalno-operatornye volterrovy uravneniya v teorii optimalnogo upravleniya raspredelennymi sistemami”, Doklady Akademii nauk, 305:5 (1989), 1056–1059 | Zbl

[2] V. I. Sumin, “Equiquasinilpotency: definitions, conditions, examples of application”, Tambov University Reports. Series: Natural and Technical Sciences, 15:1 (2010), 453–466 (In Russian)

[3] V. I. Sumin, “Ob obosnovanii gradientnykh metodov dlya raspredelennykh zadach optimalnogo upravleniya”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 30:1 (1990), 3–21 | MR | Zbl

[4] V. I. Sumin, “O dostatochnykh usloviyakh ustoichivosti suschestvovaniya globalnykh reshenii upravlyaemykh kraevykh zadach”, Differentsialnye uravneniya, 26:12 (1990), 2097–2109 | MR | Zbl

[5] V. I. Sumin, Funktsional'nye Volterrovy Uravneniya v Teorii Optimal'nogo Upravleniya Raspredelennymi Sistemami, Nizhny Novgorod State University Publ., Nizhny Novgorod, 1992 (In Russian)

[6] V. I. Sumin, “The problem of sustainability of existence global solutions of controlled boundary value problems and Volterra functional equations”, Nizhny Novgorod University Reports. Series: Mathematics, 2003, no. 1, 91–107 (In Russian) | Zbl

[7] I. V. Lisachenko, V. I. Sumin, “Nelineinaya upravlyaemaya zadacha Gursa-Darbu: usloviya sokhraneniya globalnoi razreshimosti”, Differentsialnye uravneniya, 47:6 (2011), 858–870 | MR | Zbl

[8] V. I. Sumin, A. V. Chernov, “Volterra functional-operator equations in the theory of optimization of distributed systems”, Systems Dinamics and Control Processes (SDCP'2014), International Conference “Systems Dinamics and Control Processes”, dedicated to the 90th Anniversary of the birth of Acad. N.N. Krasovskiy, Proceedings International Conference (Ekaterinburg, 2014), 2015, 293–300 (In Russian)

[9] V. Sumin, “Volterra Functional-Operator Equations in the Theory of Optimal Control of Distributed Systems”, IFAC PapersOnLine, 51:32 (2018), 759–764 | DOI

[10] A. V. Chernov, “O sokhranenii razreshimosti polulineinogo uravneniya globalnoi elektricheskoi tsepi”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 58:12 (2018), 2095–2111 | Zbl

[11] V. I. Sumin, “Controlled Volterra functional equations and the contraction mapping principle”, Trudy Instituta Matematiki I Mekhaniki URO RAN, 25:1 (2019), 262–278 (In Russian) | MR

[12] V. I. Sumin, “Controlled functional Volterra equations in Lebesgue space”, Nizhny Novgorod University Reports. Series: Mathematical Modeling and Optimal Control, 1998, no. 2(19), 138–151 (In Russian)

[13] V. I. Sumin, On Controlled Functional Volterra Equations in Lebesgue spaces, Deposited in VINITI 03.09.98, No 2742-B98, 92 pp. (In Russian)

[14] G. C. Rota, G. Strang, “A note on the joint spectral radius”, Indag. Math., 22 (1960), 379–381 | DOI | MR

[15] V. S. Shulman, Y. V. Turovskii, “Joint Spectral Radius, operator semigroups, and a problem of W. Wojtynski”, Journal of Functional Analysis, 177:2 (2000), 383–441 | DOI | MR | Zbl

[16] YU. L. Daletskiy , M. G. Kreyn, Ustoychivost' Resheniy Differentsial'nykh Uravneniy v Banakhovom Prostranstve, Nauka Publ., Moscow, 1970 (In Russian)