Solving optimal control problems for delayed control-affine systems with quadratic cost by numerical continuation

In this paper we introduce a new method to solve fixed-delay optimal control problems which exploits numerical homotopy procedures. It is known that solving this kind of problems via indirect methods is complex and computationally demanding because their implementation is faced with two difficulties: the extremal equations are of mixed type, and besides, the shooting method has to be carefully initialized. Here, starting from the solution of the non-delayed version of the optimal control problem, the delay is introduced by numerical homotopy methods. Convergence results, which ensure the effectiveness of the whole procedure, are provided. The numerical efficiency is illustrated on an example.

[1]  M. Malek-Zavarei,et al.  Time-Delay Systems: Analysis, Optimization and Applications , 1987 .

[2]  Harvey Thomas Banks.  Approximation of nonlinear functional differential equation control systems , 1979 .

[3]  M. Filomena Teodoro,et al.  The numerical solution of forward-backward differential equations: Decomposition and related issues , 2010, J. Comput. Appl. Math..

[4]  Cristiana J. Silva,et al.  Smooth Regularization of Bang-Bang Optimal Control Problems , 2010, IEEE Transactions on Automatic Control.

[5]  Emmanuel Trélat,et al.  Optimal Control and Applications to Aerospace: Some Results and Challenges , 2012, Journal of Optimization Theory and Applications.

[6]  Gamal N. Elnagar,et al.  Numerical solution of time-delayed functional differential equation control systems , 2001 .

[7]  K. H. Wong,et al.  Semi-infinite programming approach to nonlinear time-delayed optimal control problems with linear continuous constraints , 2006, Optim. Methods Softw..

[8]  H. Maurer,et al.  Optimal control problems with delays in state and control variables subject to mixed control–state constraints , 2009 .

[9]  Aldo Rustichini,et al.  Functional differential equations of mixed type: The linear autonomous case , 1989 .

[10]  K. Kaji,et al.  Nonlinearly constrained time-delayed optimal control problems , 1994 .

[11]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[12]  Emmanuel Trélat,et al.  Some Properties of the Value Function and Its Level Sets for Affine Control Systems with Quadratic Cost , 2000, math/0607424.

[13]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[14]  Emmanuel Trélat,et al.  Convergence Results for Smooth Regularizations of Hybrid Nonlinear Optimal Control Problems , 2011, SIAM J. Control. Optim..

[15]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[16]  John Mallet-Paret,et al.  The Fredholm Alternative for Functional Differential Equations of Mixed Type , 1999 .

[17]  Cheng-Liang Chen,et al.  Numerical solution of time-delayed optimal control problems by iterative dynamic programming , 2000 .