Abstract
We consider the problem of stabilizing a system of ordinary differential equations nonlinear in the state variables with control parameters. Strict pointwise constraints are imposed on the feasible values of the feedback control. Assuming sufficient smoothness of the functions on the right-hand side in the differential equations, a piecewise affine system is constructed that approximates the original nonlinear system on a rectangular grid in a given state space domain. The stabilizer can also be found in a piecewise affine form; it corresponds to a continuous but not everywhere differentiable Lyapunov function of a similar structure. The main theorem on sufficient conditions for system stabilizability with the help of piecewise affine control is stated and proved. An algorithm for constructing such a control and a Lyapunov function in a small neighborhood of the zero equilibrium is proposed. An example of numerical solution of the stabilization problem for a specific model system in three-dimensional space is considered.
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Notes
The symbol “\(\preceq \)” corresponds to the componentwise inequality for vectors of the same length.
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Funding
The results in Secs. 1–4 were obtained with the financial support of the Russian Science Foundation, project no. 22-11-00042. The results in Sec. 5 were obtained with the support of the Ministry of Science and Higher Education of the Russian Federation as part of the implementation of the program of the Moscow Center for Fundamental and Applied Mathematics under agreement no. 075-15-2022-284.
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Translated by V. Potapchouck
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Tochilin, P.A. On Constructing a Piecewise Affine Stabilizer for a Nonlinear System. Diff Equat 58, 1538–1548 (2022). https://doi.org/10.1134/S0012266122011009X
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DOI: https://doi.org/10.1134/S0012266122011009X