Solution of generalized matrix Riccati differential equation for indefinite stochastic linear quadratic singular system using neural networks.

*(English)*Zbl 1157.65397Summary: The solution of a generalized matrix Riccati differential equation (GMRDE) for an indefinite stochastic linear quadratic singular system is obtained using neural networks. The goal is to provide an optimal control with reduced calculus effort by comparing the solutions of GMRDE obtained from the well known traditional Runge Kutta (RK) method and a nontraditional neural network method.

To obtain the optimal control, the solution of the GMRDE is computed by a feed forward neural network (FFNN). The accuracy of the solution of the neural network approach to the problem is qualitatively better. The neural network solution is also compared with the solution of ode45, a standard solver available in MATLAB which implements the RK method for a variable step size.

The advantage of the proposed approach is that, once the network is trained, it allows an instantaneous evaluation of the solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method.

To obtain the optimal control, the solution of the GMRDE is computed by a feed forward neural network (FFNN). The accuracy of the solution of the neural network approach to the problem is qualitatively better. The neural network solution is also compared with the solution of ode45, a standard solver available in MATLAB which implements the RK method for a variable step size.

The advantage of the proposed approach is that, once the network is trained, it allows an instantaneous evaluation of the solution at any desired number of points spending negligible computing time and memory. The computation time of the proposed method is shorter than the traditional RK method. An illustrative numerical example is presented for the proposed method.

##### MSC:

65K10 | Numerical optimization and variational techniques |

49N10 | Linear-quadratic optimal control problems |

49J55 | Existence of optimal solutions to problems involving randomness |

65C30 | Numerical solutions to stochastic differential and integral equations |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

##### Keywords:

generalized matrix Riccati differential equation; indefinite stochastic linear singular system; neural networks; optimal control; Runge Kutta method; numerical example##### Software:

Matlab
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\textit{P. Balasubramaniam} and \textit{N. Kumaresan}, Appl. Math. Comput. 204, No. 2, 671--679 (2008; Zbl 1157.65397)

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