Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F16%3A00303396" target="_blank" >RIV/68407700:21230/16:00303396 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1002/oca.2190" target="_blank" >http://dx.doi.org/10.1002/oca.2190</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1002/oca.2190" target="_blank" >10.1002/oca.2190</a>
Alternative languages
Result language
angličtina
Original language name
Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BC - Theory and management systems
OECD FORD branch
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Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2016
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
OPTIMAL CONTROL APPLICATIONS & METHODS
ISSN
0143-2087
e-ISSN
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Volume of the periodical
37
Issue of the periodical within the volume
4
Country of publishing house
US - UNITED STATES
Number of pages
16
Pages from-to
749-764
UT code for WoS article
000379934600011
EID of the result in the Scopus database
2-s2.0-84977604879