Quadratic Optimal Control Problems with Shooting Method
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Date
2022-11-25
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Addis Ababa University
Abstract
Control theory is an area of applied mathematics that deals with principles,laws,
and desire of dynamic systems. Optimal control problems are generalized
form of variation problems. A very important tool in variational calculus is
the notion of Gateaux-differentiability. It is the basis of the development of
necessary optimality conditions.
ELDE is a necessary optimality condition to solve variational problems.
The solution of ELDE is an extremal function of a variational problem.
Characterizing theorem of convex optimization is the necessary and sufficient
condition of many convex problems. i.e Let (P) be given, S is convex set, f is
convex function and x0 ∈ S. Then x0 ∈ M(f, S) if and only if f′(x0, x−x0) ≥
0, ∀x ∈ S. In an OCP our aim is to find the optimal state function x∗(t) and
the optimal control function u∗(t) which optimize the objective functional
in t ∈ [a, b] by using necessary and sufficient optimality conditions. The
necessary optimality conditions for (x∗(t), u∗(t)) to be extremal solutions of
optimal control problem is the validity of :- Pontryagin minimum principle, of
ELDE with TR, and ODE conditions. To determine whether the extremals
are optimal solutions of OCP or not; sufficient optimality conditions are
required; (e.g checking the convexity of the objective functional and the
convexity of the feasible set). QOCP is a non linear optimization where the
cost function is quadratic but the differential equation is linear. In quadratic
control problem since the objective function is convex then the extremals
are the optimal solution of the problem. Linear QOCP is an important
type of quadratic control problem that simples the work of feed back control
system. OCP can be solved by different methods depending on the type
of the problem. This paper mainly considers solving QOCP by using the
method of lagrange multiplier and shooting method.
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Keywords
Quadratic Optimal, Control Problems, Shooting Method