A.V. Arguchintsev, V.A. Srochko. Solution of a linear–quadratic problem on a set of piecewise constant controls with parametrization of the functional ... P. 5-16

A linear–quadratic problem of optimal control with arbitrary matrices in the cost functional and a multidimensional control constrained at every time is considered. The set of admissible controls consists of piecewise constant vector functions relative to a nonuniform discretization grid. The optimal control problem is reduced to a finite-dimensional form with the use of characteristic functions with grid structure and block matrices together with the corresponding operation of scalar product. Nonnegative parameters of the quadratic forms provide the possibility of regularization of the cost functional. The choice of these parameters is aimed at the regularization of the functional in the sense of its reduction to a convex or concave structure at the level of a finite-dimensional model. The conditions for these parameters are of spectral nature; they are inequalities with respect to extreme eigenvalues of the block matrices that form the objective function. The corresponding convex or concave optimization problems allow to solve the problem in a finite number of iterations. A nongradient condition of global optimality is obtained for the original problem of optimal control based on known estimates for the increment of the functional. A nonlocal refinement procedure in terms of Pontryagin’s function is proposed.

Keywords: linear–quadratic problem, multidimensional discrete control, functional with parameters, reduction to a finite-dimensional model, regularization of the problem

Received May 30, 2022

Revised July, 5 2022

Accepted July 11, 2022

Funding Agency: This work was supported by the Vladimir Potanin Foundation (grant no. GSAD-0022/212).

Alexander Valeryevich Arguchintsev, Dr. Phys.-Math. Sci., Prof., Irkutsk State University, Irkutsk, 664003 Russia, e-mail: arguch@math.isu.ru

Vladimir Andreevich Srochko, Dr. Phys.-Math. Sci., Prof., Irkutsk State University, Irkutsk, 664003 Russia, e-mail: srochko@math.isu.ru


1.   Rao A.V. A survey of numerical methods for optimal control. Adv. Astron. Sci., 2009, vol. 135, art. no. AAS 09-334, 32 p.

2.   Golfetto W.A., da Silva Fernandes S. A review of gradient algorithms for numerical computation of optimal trajectories. J. Aerosp. Technol. Manag., 2012, vol. 4, pp. 131–143. doi: 10.5028/JATM.2012.04020512 

3.   Srochko V.A. Iteratsionnye metody resheniya zadach optimal’nogo upravleniya [Iterative methods for solving optimal control problems]. Moscow: Fizmatlit Publ., 2000, 160 p. ISBN: 5-92210086-6 .

4.   Gabasov R., Kirillova F.M., Pavlenok N.S. Constructing open-loop and closed-loop solutions of linear-quadratic optimal control problems. Comput. Math. Math. Phys., 2008, vol. 48, no. 10, pp. 1715–1745. doi: 10.1134/S0965542508100023 

5.   Kostyukova O.I., Fedartsova N.M. Investigation of solution properties of linear-quadratic parametric optimal control problem. Informatsionno-upravlyayushchie sistemy, 2012, vol. 4, pp. 43–51 (in Russian).

6.   Grad J.R., Morris K.A. Solving the linear quadratic optimal control problem for infinite-dimensional systems. Comput. Math. with Appl., 1996, vol. 32, no. 9, pp. 99–119. doi: 10.1016/0898-1221(96)00180-0 

7.   Arguchintsev A.V., Srochko V.A. Procedure for regularization of bilinear optimal control problems based on a finite-dimensional model. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2022, vol. 18, no. 1, pp. 180–188 (in Russian). doi: 10.21638/11701/spbu10.2022.115 

8.   Strekalovsky A.S. Elementy nevypukloi optimizatsii [Elements of nonconvex optimization]. Novosibirsk: Nauka Publ., 2003, 356 p. ISBN: 5-02-032064-1 .

9.   Strekalovsky A.S. On global maximum of a convex terminal functional in optimal control problems. J. Glob. Optim, 1995, vol. 7, no. 1, pp. 75–91. doi: 10.1007/BF01100206 

10.   Horn R.A., Johnson C.R. Matrix analysis. Cambridge: Cambridge University Press, 1985, 561 p. doi: 10.1017/CBO9780511810817 . Translated to Russian under the title Matrichnyi analiz. Moscow: Mir Publ., 1989, 561 p.

11.   Izmailov A.F., Solodov M.V. Chislennye metody optimizatsii [Numerical methods of optimization]. Moscow: Fizmatlit Publ., 2005, 304 p. ISBN: 5-9221-0045-9 .

12.   Pshenichny B.N., Danilin Yu.M. Numerical methods in extremal problems. Moscow: Mir Publ., 1978, 276 p. ISBN: 9781399867313 . Original Russian text published in Pshenichnyi B.N., Danilin Yu.M. Chislennye metody v ekstremal’nykh zadachakh, Moscow: Nauka Publ., 1975, 319 p.

13.   Konnov I.V. Nelineinaya optimizatsiya i variatsionnye neravenstva. [Nonlinear optimization and variational inequalities]. Kazan’: Kazan. Univ, 2013, 508 p. ISBN: 978-5-00019-059-3 .

14.   Srochko V.A., Aksenyushkina E.V., Antonik V.G. Resolution of a linear-quadratic optimal control problem based on finite-dimensional models. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 37, pp. 3–16 (in Russian). doi: 10.26516/1997-7670.2021.37.3 

15.   Parlett B.N. The symmetric eigenvalue problem. Philadelphia: SIAM, 1980, 422 p. ISBN: 9780898714029 . Translated to Russian under the title Simmetrichnaya problema sobstvennykh znachenii: Chislennye metody. Moscow: Mir Publ., 1983, 384 p.

16.   Subbotina N. N., Krupennikov E. A. Weak* approximations for the solution of a dynamic reconstruction problem. Trudy Instituta Matematiki i Mekhaniki UrO RAN, vol. 27, no. 2, pp. 208–220 (in Russian). doi: 10.21538/0134-4889-2021-27-2-208-220 .

Cite this article as: A.V. Arguchintsev, V.A. Srochko. Solution of a linear–quadratic problem on a set of piecewise constant controls with parametrization of the functional. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 3, pp. 5–16;  Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S43–S53.