# American Institute of Mathematical Sciences

May  2019, 24(5): 2281-2292. doi: 10.3934/dcdsb.2019095

## Necessary optimality conditions for an integro-differential Bolza problem via Dubovitskii-Milyutin method

 1 Faculty of Mathematics and Computer Science, University of Lodz, 90-238 Lodz, Banacha 22, Poland 2 State School of Higher Vocational Education, 96-100 Skierniewice, Batorego 64c, Poland

* Corresponding author: Stanisław Walczak

A tribute to Helmut Maurer, Urszula Ledzewicz and Heinz Schättler.
The second author has a great satisfaction to be a supervisor of the doctoral dissertation of Professor Urszula Ledzewicz

Received  December 2017 Revised  January 2019 Published  May 2019 Early access  March 2019

In the paper, we derive a maximum principle for a Bolza problem described by an integro-differential equation of Volterra type. We use the Dubovitskii-Milyutin approach.

Citation: Dariusz Idczak, Stanisław Walczak. Necessary optimality conditions for an integro-differential Bolza problem via Dubovitskii-Milyutin method. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2281-2292. doi: 10.3934/dcdsb.2019095
##### References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [2] M. Ya. Dubovitskii and A. A. Milyutin, The extremum problem in the presence of constraints, Dokl. Acad. Nauk SSSR, 149 (1963), 759-762. [3] M. Ya. Dubovitskii and A. A. Milyutin, Extremum problems in the presence of constraints, Zh. Vychisl. Mat. i Mat. Fiz., 5 (1965), 395-453. [4] I. W. Girsanow, Lectures on Mathematical Theory of Extremum Problems, Springer, New York, 1972. [5] D. Idczak and S. Walczak, Application of a global implicit function theorem to a general fractional integro-differential system of Volterra type, Journal of Integral Equations and Applications, 27 (2015), 521-554. [6] D. Idczak, Optimal control of a coercive Dirichlet problem, SIAM J. Control Optim., 36 (1998), 1250-1267.  doi: 10.1137/S0363012997296341. [7] D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces, Advanced Nonlinear Studies, 12 (2012), 89-100.  doi: 10.1515/ans-2012-0105. [8] D. Idczak, A. Skowron and S. Walczak, Sensitivity of a fractional integrodifferential Cauchy problem of Volterra type, Abstract and Applied Analysis, 2013 (2013), Article Id 129478, 8 pages. doi: 10.1155/2013/129478. [9] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems, North-Holland, 1979. [10] U. Ledzewicz, A necessary condition for a problem of optimal control with equality and inequality constraints, Control and Cybernetics, 14 (1985), 351-360. [11] U. Ledzewicz, On some specification of the Dubovitskii-Milyutin method, Nonlinear Analysis: theory, methods and Applications, 10 (1986), 1367-1371.  doi: 10.1016/0362-546X(86)90107-0. [12] U. Ledzewicz, Application of the method of contractor directions to the Dubovitskii-Milyutin formalism, Journal of Mathematical Analysis and Applications, 125 (1987), 174-184.  doi: 10.1016/0022-247X(87)90172-7. [13] U. Ledzewicz, Application of some specitfication of the Dubovitskii-Milyutin method to problems of optimal control, Nonlinear Analysis: Theory, Methods and Applications, 10 (1986), 1367-1371.  doi: 10.1016/0362-546X(86)90107-0. [14] U. Ledzewicz, H. Schättler and S. Walczak, Stability of elliptic optimal control problems, Comput. Math. Appl., 41 (2001), 1245-1256.  doi: 10.1016/S0898-1221(01)00095-5. [15] V. Volterra, Sulle equazioni integro-differenziali, R. C. Acad. Lincei (5), 18 (1909), 167-174. [16] V. Volterra, Sulle equazioni della elettrodinamica, R. C. Acad. Lincei (5), 18 (1909), 203-211. [17] V. Volterra, Sulle equazioni integro-differenziali della teoria dell’elasticita, R. C. Acad. Lincei (5), 18 (1909), 296-301. [18] V. Volterra, Equazioni integro-differenziali della elasticita nel caso della isotropia, R. C. Acad. Lincei (5), 18 (1909), 577-586.

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##### References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [2] M. Ya. Dubovitskii and A. A. Milyutin, The extremum problem in the presence of constraints, Dokl. Acad. Nauk SSSR, 149 (1963), 759-762. [3] M. Ya. Dubovitskii and A. A. Milyutin, Extremum problems in the presence of constraints, Zh. Vychisl. Mat. i Mat. Fiz., 5 (1965), 395-453. [4] I. W. Girsanow, Lectures on Mathematical Theory of Extremum Problems, Springer, New York, 1972. [5] D. Idczak and S. Walczak, Application of a global implicit function theorem to a general fractional integro-differential system of Volterra type, Journal of Integral Equations and Applications, 27 (2015), 521-554. [6] D. Idczak, Optimal control of a coercive Dirichlet problem, SIAM J. Control Optim., 36 (1998), 1250-1267.  doi: 10.1137/S0363012997296341. [7] D. Idczak, A. Skowron and S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces, Advanced Nonlinear Studies, 12 (2012), 89-100.  doi: 10.1515/ans-2012-0105. [8] D. Idczak, A. Skowron and S. Walczak, Sensitivity of a fractional integrodifferential Cauchy problem of Volterra type, Abstract and Applied Analysis, 2013 (2013), Article Id 129478, 8 pages. doi: 10.1155/2013/129478. [9] A. D. Ioffe and V. M. Tikhomirov, Theory of Extremum Problems, North-Holland, 1979. [10] U. Ledzewicz, A necessary condition for a problem of optimal control with equality and inequality constraints, Control and Cybernetics, 14 (1985), 351-360. [11] U. Ledzewicz, On some specification of the Dubovitskii-Milyutin method, Nonlinear Analysis: theory, methods and Applications, 10 (1986), 1367-1371.  doi: 10.1016/0362-546X(86)90107-0. [12] U. Ledzewicz, Application of the method of contractor directions to the Dubovitskii-Milyutin formalism, Journal of Mathematical Analysis and Applications, 125 (1987), 174-184.  doi: 10.1016/0022-247X(87)90172-7. [13] U. Ledzewicz, Application of some specitfication of the Dubovitskii-Milyutin method to problems of optimal control, Nonlinear Analysis: Theory, Methods and Applications, 10 (1986), 1367-1371.  doi: 10.1016/0362-546X(86)90107-0. [14] U. Ledzewicz, H. Schättler and S. Walczak, Stability of elliptic optimal control problems, Comput. Math. Appl., 41 (2001), 1245-1256.  doi: 10.1016/S0898-1221(01)00095-5. [15] V. Volterra, Sulle equazioni integro-differenziali, R. C. Acad. Lincei (5), 18 (1909), 167-174. [16] V. Volterra, Sulle equazioni della elettrodinamica, R. C. Acad. Lincei (5), 18 (1909), 203-211. [17] V. Volterra, Sulle equazioni integro-differenziali della teoria dell’elasticita, R. C. Acad. Lincei (5), 18 (1909), 296-301. [18] V. Volterra, Equazioni integro-differenziali della elasticita nel caso della isotropia, R. C. Acad. Lincei (5), 18 (1909), 577-586.
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