Geodesic
General scheme for constructing the determining function in a control problem for a dynamical system with partial derivatives of different orders
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 78-88

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For a control system with partial derivatives, a criterion for the complete controllability is derived by using the cascade decomposition method based on the transition from the original system to reduced systems in subspaces. We obtain a function, which belongs to a subspace of minimal dimension and determines the type of solution of the program control problem, i.e., the state and control functions in the analytical form. Necessary and sufficient conditions for the existence of the determining function are established and a scheme of its construction is given. Necessary and sufficient conditions for the existence of a determining function in the polynomial, exponential, and fractional-rational forms are found; formulas for constructing such functions are proposed. For the original system, a solution of the program control problem is constructed.
Keywords: dynamical system with partial derivatives, complete controllability, program control, cascade decomposition method 
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     author = {E. V. Raetskaya},
     title = {General scheme for constructing the determining function in a control problem for a dynamical system with partial derivatives of different orders},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {78--88},
     publisher = {mathdoc},
     volume = {232},
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     url = {http://geodesic.mathdoc.fr/item/INTO_2024_232_a6/}
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E. V. Raetskaya. General scheme for constructing the determining function in a control problem for a dynamical system with partial derivatives of different orders. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 3, Tome 232 (2024), pp. 78-88. http://geodesic.mathdoc.fr/item/INTO_2024_232_a6/