Discrete-time optimal control problems with time delay argument: New discrete-time Euler-Lagrange equations with...

Area of Science:

• Control Theory
• Applied Mathematics
• Dynamic Systems

Background:

• Optimal control problems are essential for managing systems that change over time.
• State delays in discrete-time systems present significant challenges in control strategy development.
• Existing methods may not efficiently handle linear systems with state delays across different time horizons.

Purpose of the Study:

• To introduce a novel technique for solving linear discrete-time optimal control problems with state delays.
• To ensure system stability and optimize control strategies for both finite and infinite time horizons.
• To provide a practical and broadly applicable method for complex control issues.

Main Methods:

• Utilizing a Riccati matrix equation to optimize control strategies.
• Adopting a Bolza problem formulation for the performance index.
• Employing Euler-Lagrange equations and Pontryagin's maximum principle to simplify problems.

Main Results:

• The proposed method effectively solves linear discrete-time optimal control problems with state delays.
• System stability is ensured through bounded control inputs derived from the Riccati equation.
• The technique simplifies complex problems into manageable Riccati matrix equations.

Conclusions:

• The developed technique offers a robust solution for optimal control problems with state delays.
• Validation with examples from quantum and classical physics demonstrates its practical utility.
• This approach has potential for wide-ranging applications in various scientific and engineering fields.