On controlled Hamilton and Hamilton-Jacobi differential equations of higher-order

Area of Science:

• Nonlinear dynamics
• Theoretical mechanics
• Mathematical physics

Background:

• Controlled Lagrangians are fundamental in classical and analytical mechanics.
• Higher-order derivatives introduce complexities in dynamical system modeling.
• Existing frameworks often focus on first-order derivatives, limiting applicability.

Purpose of the Study:

• To investigate the nonlinear dynamics of controlled Lagrangians with higher-order derivatives.
• To formulate and derive the governing Hamilton-Jacobi equations for these systems.
• To establish and prove an invariance property related to the state variable.

Main Methods:

• Development of controlled higher-order Hamilton ordinary differential equations (ODEs).
• Formulation of the corresponding Hamilton-Jacobi partial differential equation (PDE).
• Mathematical proof of an invariance result with respect to the state variable.

Main Results:

• The controlled higher-order Hamilton ODEs and Hamilton-Jacobi PDE were successfully established.
• A novel invariance result concerning the state variable was formulated and proven.
• Theoretical findings were validated through illustrative applications.

Conclusions:

• The study provides a robust theoretical framework for analyzing nonlinear dynamics in higher-order controlled Lagrangians.
• The derived Hamilton-Jacobi equations and invariance property offer new tools for system analysis and control.
• The presented applications demonstrate the practical utility and effectiveness of the developed methodology.