Analytic Solutions to Optimal Control Problems with Constraints

Author(s)
Dan Wu

Affiliation(s)

Department of Mathematics, Henan University of Science and Technology, Luoyang, China.

Department of Mathematics, Henan University of Science and Technology, Luoyang, China.

ABSTRACT

In this paper, the analytic solutions to constrained optimal control problems are considered. A novel approach based on canonical duality theory is developed to derive the analytic solution of this problem by reformulating a constrained optimal control problem into a global optimization problem. A differential flow is presented to deduce some optimality conditions for solving global optimizations, which can be considered as an extension and a supplement of the previous results in canonical duality theory. Some examples are given to illustrate the applicability of our results.

In this paper, the analytic solutions to constrained optimal control problems are considered. A novel approach based on canonical duality theory is developed to derive the analytic solution of this problem by reformulating a constrained optimal control problem into a global optimization problem. A differential flow is presented to deduce some optimality conditions for solving global optimizations, which can be considered as an extension and a supplement of the previous results in canonical duality theory. Some examples are given to illustrate the applicability of our results.

KEYWORDS

Constrained Optimal Control, Analytic Solution, Canonical Duality Theory, Global Optimization

Constrained Optimal Control, Analytic Solution, Canonical Duality Theory, Global Optimization

Cite this paper

Wu, D. (2015) Analytic Solutions to Optimal Control Problems with Constraints.*Applied Mathematics*, **6**, 2326-2339. doi: 10.4236/am.2015.614205.

Wu, D. (2015) Analytic Solutions to Optimal Control Problems with Constraints.

References

[1] Casti, J. (1980) The Linear-Quadratic Control Problem: Some Recent Results and Outstanding Problems. SIAM Review, 22, 459-485.

http://dx.doi.org/10.1137/1022089

[2] Robinson, C. (1995) Dynamical Systems. CRC Press, London.

[3] Anderson, B.D.O. and Moore, J.B. (1971) Linear Optimal Control. Prentice-Hall, New Jersey.

[4] Heinkenschloss, M. and Tr?ltzsch. F. (1999) Analysis of the Lagrange-SQP-Newton Method for the Control of a Phase Field Equation. Control and Cybernetics, 28, 177-211.

[5] Kunisch, K. and Sachs, E.W. (1992) Reduced SQP Methods for Parameter Identification Problems. SIAM Journal on Numerical Analysis, 29, 1793-1820.

http://dx.doi.org/10.1137/0729100

[6] Tröltzsch, F. (1994) An SQP-Method for Optimal Control of a Nonlinear Heat Equation. Control and Cybernetics, 23, 268-288.

[7] Tian, T. and Dunn, J.C. (1994) On the Gradient Projection Method for Optimal Control Problems with Nonnegative L2 inputs. SIAM Journal on Control and Optimization, 32, 516-537.

[8] Kelley, C.T. and Sachs, E.W. (1995) Solution of Optimal Control Problems by a Pointwise Projected Newton Method. SIAM Journal on Control and Optimization, 33, 1731-1757.

http://dx.doi.org/10.1137/S0363012993249900

[9] Gao, D.Y., Ruan, N. and Latorre, V. (2014) Canonical Duality-Triality Theory: Bridge between Nonconvex Analysis/ Mechanics and Global Optimization in Complex Systems. Mathematics and Mechanics of Solids, 12, 716-735.

[10] Gao, D.Y. and Ruan, N. (2015) Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems. Mathematics and Mechanics of Solids.

http://dx.doi.org/10.1177/1081286515591087

[11] Latorre, V. and Sagratella, S. (2014) A Canonical Duality Approach for the Solution of Affine Quasi-Variational Inequalities. Journal of Global Optimization, 1, 1-17.

[12] Gao, D.Y. and Ruan, N. (2015) Application of Canonical Duality Theory to Fixed Point Problem. Springer Proceedings in Mathematics & Statistics, 95, 157-163.

http://dx.doi.org/10.1007/978-3-319-08377-3_17

[13] Zhu, J., Tao, S. and Gao, D.Y. (2009) A Study on Concave Optimization via Canonical Dual Function. Journal of Computational and Applied Mathematics, 224, 459-464.

http://dx.doi.org/10.1016/j.cam.2008.05.011

[14] Zhu, J. and Yan, W. (2009) Solution to Constrained Nonlinear Programming by Canonical Dual Method. Lecture Notes on Decision Sciences, 12, 217-222.

[15] Zhu, J., Wu, D. and Gao, D.Y. (2012) Applying the Canonical Dual Theory in Optimal Control Problems. Journal of Global Optimization, 29, 377-399.

http://dx.doi.org/10.1007/s10898-009-9474-3

[16] Adjiman, C.S., Androulakis, I.P. and Floudas, C.A. (1998) A Global Optimization Method, αBB, for General Twice-Differentiable Constrained NLPs—II. Implementation and Computational Results. Computers & Chemical Engineering, 22, 1159-1179.

http://dx.doi.org/10.1016/S0098-1354(98)00218-X

[17] Adjiman, C.S., Androulakis, I.P. and Floudas, C.A. (1998) A Global Optimization Method, αBB, for General Twice-Differentiable Constrained NLPs—I. Theoretical Advances. Computers & Chemical Engineering, 22, 1137-1158.

http://dx.doi.org/10.1016/S0098-1354(98)00027-1

[18] Keller, H.B. (1976) Numerical Solution of Two Point Boundary Value Problems. SIAM, Philadelphia.

http://dx.doi.org/10.1137/1.9781611970449

[19] Ascher, U.M., Mattheij, R.M.M. and Russell, R.D. (1995) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Classics in Applied Mathematics). SIAM, Philadelphia.

http://dx.doi.org/10.1137/1.9781611971231

[1] Casti, J. (1980) The Linear-Quadratic Control Problem: Some Recent Results and Outstanding Problems. SIAM Review, 22, 459-485.

http://dx.doi.org/10.1137/1022089

[2] Robinson, C. (1995) Dynamical Systems. CRC Press, London.

[3] Anderson, B.D.O. and Moore, J.B. (1971) Linear Optimal Control. Prentice-Hall, New Jersey.

[4] Heinkenschloss, M. and Tr?ltzsch. F. (1999) Analysis of the Lagrange-SQP-Newton Method for the Control of a Phase Field Equation. Control and Cybernetics, 28, 177-211.

[5] Kunisch, K. and Sachs, E.W. (1992) Reduced SQP Methods for Parameter Identification Problems. SIAM Journal on Numerical Analysis, 29, 1793-1820.

http://dx.doi.org/10.1137/0729100

[6] Tröltzsch, F. (1994) An SQP-Method for Optimal Control of a Nonlinear Heat Equation. Control and Cybernetics, 23, 268-288.

[7] Tian, T. and Dunn, J.C. (1994) On the Gradient Projection Method for Optimal Control Problems with Nonnegative L2 inputs. SIAM Journal on Control and Optimization, 32, 516-537.

[8] Kelley, C.T. and Sachs, E.W. (1995) Solution of Optimal Control Problems by a Pointwise Projected Newton Method. SIAM Journal on Control and Optimization, 33, 1731-1757.

http://dx.doi.org/10.1137/S0363012993249900

[9] Gao, D.Y., Ruan, N. and Latorre, V. (2014) Canonical Duality-Triality Theory: Bridge between Nonconvex Analysis/ Mechanics and Global Optimization in Complex Systems. Mathematics and Mechanics of Solids, 12, 716-735.

[10] Gao, D.Y. and Ruan, N. (2015) Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems. Mathematics and Mechanics of Solids.

http://dx.doi.org/10.1177/1081286515591087

[11] Latorre, V. and Sagratella, S. (2014) A Canonical Duality Approach for the Solution of Affine Quasi-Variational Inequalities. Journal of Global Optimization, 1, 1-17.

[12] Gao, D.Y. and Ruan, N. (2015) Application of Canonical Duality Theory to Fixed Point Problem. Springer Proceedings in Mathematics & Statistics, 95, 157-163.

http://dx.doi.org/10.1007/978-3-319-08377-3_17

[13] Zhu, J., Tao, S. and Gao, D.Y. (2009) A Study on Concave Optimization via Canonical Dual Function. Journal of Computational and Applied Mathematics, 224, 459-464.

http://dx.doi.org/10.1016/j.cam.2008.05.011

[14] Zhu, J. and Yan, W. (2009) Solution to Constrained Nonlinear Programming by Canonical Dual Method. Lecture Notes on Decision Sciences, 12, 217-222.

[15] Zhu, J., Wu, D. and Gao, D.Y. (2012) Applying the Canonical Dual Theory in Optimal Control Problems. Journal of Global Optimization, 29, 377-399.

http://dx.doi.org/10.1007/s10898-009-9474-3

[16] Adjiman, C.S., Androulakis, I.P. and Floudas, C.A. (1998) A Global Optimization Method, αBB, for General Twice-Differentiable Constrained NLPs—II. Implementation and Computational Results. Computers & Chemical Engineering, 22, 1159-1179.

http://dx.doi.org/10.1016/S0098-1354(98)00218-X

[17] Adjiman, C.S., Androulakis, I.P. and Floudas, C.A. (1998) A Global Optimization Method, αBB, for General Twice-Differentiable Constrained NLPs—I. Theoretical Advances. Computers & Chemical Engineering, 22, 1137-1158.

http://dx.doi.org/10.1016/S0098-1354(98)00027-1

[18] Keller, H.B. (1976) Numerical Solution of Two Point Boundary Value Problems. SIAM, Philadelphia.

http://dx.doi.org/10.1137/1.9781611970449

[19] Ascher, U.M., Mattheij, R.M.M. and Russell, R.D. (1995) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (Classics in Applied Mathematics). SIAM, Philadelphia.

http://dx.doi.org/10.1137/1.9781611971231