Affiliation(s)

^{1}
Faculty of Economics and Business Administration, Sofia University “St. Kliment Ohridski”, Sofia, Bulgaria.

^{2}
Pedagogical College Dobrich, Shoumen University, Shoumen, Bulgaria.

ABSTRACT

We consider a set of continuous algebraic Riccati equations with indefinite quadratic parts that arise in H￥ control problems. It is well known that the approach for solving such type of equations is proposed in the literature. Two matrix sequences are constructed. Three effective methods are described for computing the matrices of the second sequence, where each matrix is the stabilizing solution of the set of Riccati equations with definite quadratic parts. The acceleration modifications of the described methods are presented and applied. Computer realizations of the presented methods are numerically compared. In addition, a second iterative method is proposed. It constructs one matrix sequence which converges to the stabilizing solution to the given set of Riccati equations with indefinite quadratic parts. The convergence properties of the second method are commented. The iterative methods are numerically compared and investigated.

KEYWORDS

*H*_{∞} Optimal Control Problem,
Generalized Riccati Equation,
Indefinite Sign,
Stabilizing Solution

Cite this paper

G. Ivanov, I. and Ivanov, I. (2015)*H*_{∞} Optimal Control Problems for Jump. *Journal of Mathematical Finance*, **5**, 337-347. doi: 10.4236/jmf.2015.54029.

G. Ivanov, I. and Ivanov, I. (2015)

References

[1] Lanzon, A., Feng, Y., Anderson, B. and Rotkowitz, M. (2008) Computing the Positive Stabilizing Solution to Algebraic Riccati Equations with an Indefinite Quadratic Term via a Recursive Method. IEEE Transactions on Automatic Control, 53, 2280-2291.

http://dx.doi.org/10.1109/TAC.2008.2006108

[2] Dragan, V., Freiling, G., Morozan, T. and Stoica, A.-M. (2008) Iterative Algorithms for Stabilizing Solutions of Game Theoretic Riccati Equations of Stochastic Control. Proceedings of the 18th International Symposium on Mathematical Theory of Networks & Systems, Blacksburg, Virginia, 28 July-1 August 2008, 1-11.

http://scholar.lib.vt.edu/MTNS/Papers/078.pdf

[3] Vrabie, D. and Lewis, F. (2011) Adaptive Dynamic Programming for Online Solution of a Zero-Sum Differential Game. Journal of Control Theory and Applications, 9, 353-360.

http://dx.doi.org/10.1007/s11768-011-0166-4

[4] Praveen, P. and Bhasin, S. (2013) Online Partially Model-Free Solution of Two-Player Zero Sum Differential Games. 10th IFAC International Symposium on Dynamics and Control of Process Systems, India, 18-20 December 2013, 696-701.

[5] Feng, Y.T. and Anderson, B.D.O. (2010) An Iterative Algorithm to Solve State-Perturbed Stochastic Algebraic Riccati Equations in LQ Zero-Sum Games. Systems & Control Letters, 59, 50-56.

http://dx.doi.org/10.1016/j.sysconle.2009.11.006

[6] Dragan, V. and Ivanov, I. (2011) Computation of the Stabilizing Solution of Game Theoretic Riccati Equation Arising in Stochastic*H*_{∞} Control Problems. Numerical Algorithms, 57, 357-375.

http://dx.doi.org/10.1007/s11075-010-9432-7

[7] Hata, H. and Sekine, J. (2013) Risk-Sensitive Asset Management under a Wishart Autoregressive Factor Model. Journal of Mathematical Finance, 3, 222-229.

http://dx.doi.org/10.4236/jmf.2013.31A021

[8] Hudgins, D. and Na, J. (2013)*H*_{∞}-Optimal Control for Robust Financial Asset and Input Purchasing Decisions. Journal of Mathematical Finance, 3, 335-346.

http://dx.doi.org/10.4236/jmf.2013.33034

[9] Pang, W.-K., Ni, Y.-H., Li, X. and Yiu, K.-F. (2014) Continuous-Time Mean-Variance Portfolio Selection with Partial Information. Journal of Mathematical Finance, 4, 353-365.

http://dx.doi.org/10.4236/jmf.2014.45033

[10] Dragan, V., Morozan, T. and Stoica, A.M. (2013) Mathematical Methods in Robust Control of Linear Stochastic Systems. Springer, New York.

[11] Ivanov, I. (2008) On Some Iterations for Optimal Control of Jump Linear Equations. Nonlinear Analysis Series A: Theory, Methods & Applications, 69, 4012-4024.

[12] Rami, M. and Zhou, X. (2000) Linear Matrix Inequalities, Riccati Equations, and Indefinite Stochastic Linear Quadratic Controls. IEEE Transactions on Automatic Control, 45, 1131-1143.

http://dx.doi.org/10.1109/9.863597

[13] Ivanov, I. (2012) Accelerated LMI Solvers for the Maximal Solution to a Set of Discrete-Time Algebraic Riccati Equations. Applied Mathematics E-Notes, 12, 228-238.

[1] Lanzon, A., Feng, Y., Anderson, B. and Rotkowitz, M. (2008) Computing the Positive Stabilizing Solution to Algebraic Riccati Equations with an Indefinite Quadratic Term via a Recursive Method. IEEE Transactions on Automatic Control, 53, 2280-2291.

http://dx.doi.org/10.1109/TAC.2008.2006108

[2] Dragan, V., Freiling, G., Morozan, T. and Stoica, A.-M. (2008) Iterative Algorithms for Stabilizing Solutions of Game Theoretic Riccati Equations of Stochastic Control. Proceedings of the 18th International Symposium on Mathematical Theory of Networks & Systems, Blacksburg, Virginia, 28 July-1 August 2008, 1-11.

http://scholar.lib.vt.edu/MTNS/Papers/078.pdf

[3] Vrabie, D. and Lewis, F. (2011) Adaptive Dynamic Programming for Online Solution of a Zero-Sum Differential Game. Journal of Control Theory and Applications, 9, 353-360.

http://dx.doi.org/10.1007/s11768-011-0166-4

[4] Praveen, P. and Bhasin, S. (2013) Online Partially Model-Free Solution of Two-Player Zero Sum Differential Games. 10th IFAC International Symposium on Dynamics and Control of Process Systems, India, 18-20 December 2013, 696-701.

[5] Feng, Y.T. and Anderson, B.D.O. (2010) An Iterative Algorithm to Solve State-Perturbed Stochastic Algebraic Riccati Equations in LQ Zero-Sum Games. Systems & Control Letters, 59, 50-56.

http://dx.doi.org/10.1016/j.sysconle.2009.11.006

[6] Dragan, V. and Ivanov, I. (2011) Computation of the Stabilizing Solution of Game Theoretic Riccati Equation Arising in Stochastic

http://dx.doi.org/10.1007/s11075-010-9432-7

[7] Hata, H. and Sekine, J. (2013) Risk-Sensitive Asset Management under a Wishart Autoregressive Factor Model. Journal of Mathematical Finance, 3, 222-229.

http://dx.doi.org/10.4236/jmf.2013.31A021

[8] Hudgins, D. and Na, J. (2013)

http://dx.doi.org/10.4236/jmf.2013.33034

[9] Pang, W.-K., Ni, Y.-H., Li, X. and Yiu, K.-F. (2014) Continuous-Time Mean-Variance Portfolio Selection with Partial Information. Journal of Mathematical Finance, 4, 353-365.

http://dx.doi.org/10.4236/jmf.2014.45033

[10] Dragan, V., Morozan, T. and Stoica, A.M. (2013) Mathematical Methods in Robust Control of Linear Stochastic Systems. Springer, New York.

[11] Ivanov, I. (2008) On Some Iterations for Optimal Control of Jump Linear Equations. Nonlinear Analysis Series A: Theory, Methods & Applications, 69, 4012-4024.

[12] Rami, M. and Zhou, X. (2000) Linear Matrix Inequalities, Riccati Equations, and Indefinite Stochastic Linear Quadratic Controls. IEEE Transactions on Automatic Control, 45, 1131-1143.

http://dx.doi.org/10.1109/9.863597

[13] Ivanov, I. (2012) Accelerated LMI Solvers for the Maximal Solution to a Set of Discrete-Time Algebraic Riccati Equations. Applied Mathematics E-Notes, 12, 228-238.