×

Symmetries and analytical solutions of the Hamilton-Jacobi-Bellman equation for a class of optimal control problems. (English) Zbl 1346.49004

Summary: The main contribution of this paper is to identify explicit classes of locally controllable second-order systems and optimization functionals for which optimal control problems can be solved analytically, assuming that a differentiable optimal cost-to-go function exists for such control problems. An additional contribution of the paper is to obtain a Lyapunov function for the same classes of systems. The paper addresses the Lie point symmetries of the Hamilton-Jacobi-Bellman (HJB) equation for optimal control of second-order nonlinear control systems that are affine in a single input and for which the cost is quadratic in the input. It is shown that if there exists a dilation symmetry of the HJB equation for optimal control problems in this class, this symmetry can be used to obtain a solution. It is concluded that when the cost on the state preserves the dilation symmetry, solving the optimal control problem is reduced to finding the solution to a first-order ordinary differential equation. For some cases where the cost on the state breaks the dilation symmetry, the paper also presents an alternative method to find analytical solutions of the HJB equation corresponding to additive control inputs. The relevance of the proposed methodologies is illustrated in several examples for which analytical solutions are found, including the Van der Pol oscillator and mass-spring systems. Furthermore, it is proved that in the well-known case of a linear quadratic regulator, the quadratic cost is precisely the cost that preserves the dilation symmetry of the equation.

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
49N10 Linear-quadratic optimal control problems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bryson, Applied Optimal Control (1975)
[2] Kalman, When is a linear control system optimal?, ASME Transactions, Journal of Basic Engineering 86 pp 51– (1964) · doi:10.1115/1.3653115
[3] Lukes, Optimal regulation of nonlinear dynamical systems, SIAM Journal of Control 7 (1) pp 75– (1969) · Zbl 0184.18802 · doi:10.1137/0307007
[4] Doyle, nonlinear games: examples and counterexamples, IEEE Conference on Decision and Control, Kobe, Japan pp 3915– (1996)
[5] Freeman, Inverse optimality in robust stabilization, SIAM Journal on Control and Optimization 34 (4) pp 1365– (1996) · Zbl 0863.93075 · doi:10.1137/S0363012993258732
[6] Margaliot, Some nonlinear optimal control problems with closed-form solutions, International Journal of Robust and Nonlinear Control 11 pp 1365– (2001) · Zbl 1024.93025 · doi:10.1002/rnc.600
[7] Krstić, Inverse optimal design of input-to-state stabilizing nonlinear controllers, IEEE Transactions on Automatic Control 43 (3) pp 336– (1998) · Zbl 0910.93064 · doi:10.1109/9.661589
[8] Pan, Backstepping design with local optimality matching, IEEE Transactions on Automatic Control 46 (7) pp 1014– (2001) · Zbl 1007.93025 · doi:10.1109/9.935055
[9] Krstic, Inverse optimal stabilization of a rigid spacecraft, IEEE Transactions on Automatic Control 44 (5) pp 1042– (1999) · Zbl 1136.93424 · doi:10.1109/9.763225
[10] Guojun, Inverse optimal stabilization of a class of nonlinear systems, Proceedings of the 26th Chinese Control Conference pp 226– (2007)
[11] Kanazawa, IEEE Conference on Decision and Control, pp. 2260-2267, 2009 pp 493– (1994)
[12] Naicker, Symmetry reductions of a Hamilton-Jacobi-Bellman equation arising in financial mathematics, Journal of Nonlinear Mathematical Physics 12 (2) pp 268– (2005) · Zbl 1080.35163 · doi:10.2991/jnmp.2005.12.2.8
[13] Leach, Symmetry-based solution of a model for a combination of a risky investment and a riskless investment, Journal of Mathematical Analysis Applications 334 pp 368– (2007) · Zbl 1154.91027 · doi:10.1016/j.jmaa.2006.11.056
[14] Dimas, Complete specification of some partial differential equations that arise in financial mathematics, Journal of Nonlinear Mathematical Physics 16 pp 73– (2009) · Zbl 1362.35310 · doi:10.1142/S1402925109000339
[15] Athans, Optimal Control: An Introduction to the Theory and Its Applications (1966)
[16] Cantwell, Introduction to Symmetry Analysis (2002)
[17] Rodrigues L An inverse optimality method to solve a class of second order optimal control problems 18th Mediterranean Conference on Control & Automation Marrakech, Morocco 2010 407 412
[18] Khalil, Nonlinear Systems (1996)
[19] Omrani BG Rabbath CA Rodrigues L An inverse optimality method to solve a class of third-order optimal control problems IEEE Conference on Decision and Control Atlanta, GA, USA 2010 4845 4850
[20] Fallah MA Rodrigues L Optimal control of a third order nonlinear system based on an inverse optimality method American Control Conference San Francisco, CA, USA 2011 900 904
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.