Topics in Applied Mathematics (Calculus of Variations and Optimal Control) | MA

Brief Information

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Summary

Control Theory

The mathematical study of how to manipulate the parameters affecting the behavior of a system to produce the desired or optimal outcome. – Control theory | Wolfram MathWorld

Calculus of Variations

A field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. – Calculus of Variations | Wikipedia

Optimal Control Theory

Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. – Optimal Control | Wikipedia

Control System / Dynamic System

A control system, i.e.,  dynamic system is composed of three components: time, state, and controls. A control system is used to describe the state that varies by time and controls. The number of controls can be one or more. However, we first consider the dynamic systems with one control. The control system is written as follows in equations.

\frac{dx}{dt}=f(x(t)), u(t))
where t is time, x(f) is the state function, and u(t) is the control function.

Theorems of the Euler-Lagrange Equation

 

Theorem of the Hamiltonian Equation

H(p,x,u,t)=F(x,u,t)+pf(x,u)

 

 

 

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