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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 |
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.
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. |
show all references
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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