# Algebraic Solutions to the Hamilton-Jacobi Equation with the Time-Varying Hamiltonian

@article{Kawano2013AlgebraicST, title={Algebraic Solutions to the Hamilton-Jacobi Equation with the Time-Varying Hamiltonian}, author={Yu Kawano and Toshiyuki Ohtsuka}, journal={SICE journal of control, measurement, and system integration}, year={2013}, volume={6}, pages={28-37} }

The Hamilton-Jacobi equation (HJE) with the time-varying Hamiltonian plays an important role in the analysis and control of nonlinear systems and is very difficult to solve for general nonlinear systems. In this paper, the HJE with coefficients belonging to meromorphic functions is considered, and its solutions with algebraic gradients are characterized in terms of commutative algebra. It is shown that there exists a solution with an algebraic gradient if and only if an Hinvariant and…

## 5 Citations

A Recursive Elimination Method for Finite-Horizon Optimal Control Problems of Discrete-Time Rational Systems

- MathematicsIEEE Transactions on Automatic Control
- 2014

A method of solving finite-horizon optimal control problems involving discrete-time rational systems is proposed, whereby Sequences of algebraic equations for the control input and costate at each time are constructed backward by the recursive elimination of variables in the optimality conditions.

Title A Recursive Elimination Method for Finite-Horizon OptimalControl Problems of Discrete-Time Rational Systems

- Mathematics
- 2017

In this technical note, a method of solving finite-horizon optimal control problems involving discrete-time rational systems is proposed. Sequences of algebraic equations for the control input and…

On First Integrals of Hamiltonian System with Holonomic Hamiltonian

- MathematicsArXiv
- 2022

In this study, the solution of the Hamilton-Jacobi equation (HJE) with holonomic Hamiltonian is investigated in terms of the first integrals of the corresponding Hamiltonian system. Holonomic…

## References

SHOWING 1-10 OF 33 REFERENCES

Solutions to the Hamilton-Jacobi Equation With Algebraic Gradients

- MathematicsIEEE Transactions on Automatic Control
- 2011

The so-called curse of dimensionality can be removed when a solution to the Hamilton-Jacobi equation with coefficients consisting of rational functions with an algebraic gradient exists.

Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

- MathematicsIEEE Transactions on Automatic Control
- 2008

Two methods for approximating the stabilizing solution of the Hamilton-Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory to provide analytical approximations of the stable Lagrangian submanifold from which the stabilizer solution is derived.

Analysis of the Hamilton--Jacobi Equation in Nonlinear Control Theory by Symplectic Geometry

- MathematicsSIAM J. Control. Optim.
- 2002

In this paper, the geometric property and structure of the Hamilton--Jacobi equation arising from nonlinear control theory are investigated using symplectic geometry and it is seen that many fundamental properties of the Riccati equation can be generalized in the Hamilton-Jacobi equations.

Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations

- Mathematics
- 1997

Preface.- Basic notations.- Outline of the main ideas on a model problem.- Continuous viscosity solutions of Hamilton-Jacobi equations.- Optimal control problems with continuous value functions:…

Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations.

- Mathematics
- 1984

Abstract : Recently M. G. Crandall and P. L. Lions introduced the notion of 'viscosity solutions' of scalar nonlinear first order partial differential equations. Viscosity solutions need not be…

L/sub 2/-gain analysis of nonlinear systems and nonlinear state-feedback H/sub infinity / control

- Mathematics
- 1992

Previously obtained results on L2-gain analysis of smooth nonlinear systems are unified and extended using an approach based on Hamilton-Jacobi equations and inequalities, and their relation to…

Integrable systems in the realm of algebraic geometry

- Mathematics
- 1996

Introduction.- Integrable Hamiltonian systems on affine Poisson varietie: Affine Poisson varieties and their morphisms Integrable Hamiltonian systems and their morphisms Integrable Hamiltonian…

${\cal H}_\infty$ Control of Nonlinear Systems: Differential Games and Viscosity Solutions

- Mathematics
- 1996

Dealing with disturbances is one of the most important questions for controlled systems. ${\cal H}_\infty$ optimal control theory is a deterministic way to tackle the problem in the presence of…