P. E. Makoveeva, A. V. Egorov

St. Petersburg State University (St. Petersburg, Russian Federation)

Abstract. The Lyapunov functional method is applied to a linear parabolic-type equation with homogeneous boundary conditions. Within this framework, a Lyapunov functional is constructed whose derivative along the solutions of the system is a prescribed negative definite quadratic form. A central role in this construction is played by the Lyapunov matrix, whose properties are investigated in detail. In the paper, two definitions of the Lyapunov matrix are proposed. The first one is based on its representation in the form of a series. The second alternative definition relates the matrix to the Green’s function for a corresponding stationary equation. The consistency of the proposed definitions is established, and it is proved that any function satisfying the second definition simultaneously satisfies the first one, thereby confirming the equivalence of the two approaches. An important advantage of the second definition lies in its constructive nature: this makes it possible to derive an explicit analytical representation of the Lyapunov matrix for arbitrary parameters of the boundary value problem. Moreover, it is shown that this approach allows construction of Lyapunov functionals with a prescribed derivative without imposing the requirement of exponential stability. This significantly broadens the scope of potential applications.

Key Words: Lyapunov matrix, parabolic equation, Lyapunov functional, functional with a prescribed derivative, exponential stability

For citation: P. E. Makoveeva, A. V. Egorov. Construction of a functional with a prescribed derivative for a linear parabolic equation with homogeneous boundary conditions. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:1(2026), 31–47. DOI: https://doi.org/10.15507/2079-6900.28.202601.31-47

Information about the authors:

Polina E. Makoveeva, Postgraduate Student, Faculty of Applied Mathematics and Control Processes, St. Petersburg State University (7-9 Universitetskaya Emb., St. Petersburg 199034, Russia) ORCID: https://orcid.org/0009-0006-2846-4829, p.e.makoveeva@spbu.ru

Alexey V. Egorov, Ph. D. (Phys. and Math.), Associate Professor, Department of Control Theory, St. Petersburg State University (7-9 Universitetskaya Emb., St. Petersburg 199034, Russia), ORCID: https://orcid.org/0000-0001-7671-2467, alexey.egorov@spbu.ru