Optimization in Transition between Two Dynamic Systems Governed by a Class of Weakly Singular Integro-Differential Equations

This study presents numerical methods for solving the minimum energies that satisfy typical optimal requirements in the transition between two dynamic systems where each system is governed by a different kind of weakly singular integro-differential equation. The class of weakly singular inte-gro-differential equations originates from mathematical models in aeroelasticity. The proposed numerical methods are based on earlier reported ap-proximation schemes for the equations of the first kind and the second kind. The main result of this study is the development of numerical techniques for determining the stability between two dynamic systems in the minimum energy sense.


Introduction
The minimum energy problem and the associated optimal control problem have been investigated for more than half a century. The system constraints can be ordinary differential equations, partial differential equations, or functional differential equations. This study introduces a numerical method for finding the minimum energy to satisfy the general criterion that can be adjusted to minimize various requirements through the selection of appropriate parameters. One system constraint is the class of equations of the first kind, which originates from an aeroelasticity problem where the mathematical model consists of eight integro-differential equations [1]. In the model, the most determinate equation is a scalar weakly singular integro-differential equation of the first kind [2] [3].
Furthermore, because of the natural facts of transition between liquid water and solid ice [4] or the aviation transition between vertical take-off and horizontal flight of an unmanned aerial vehicle [5], we were interested in the energy issue in the transition between two basically different (but related) dynamic systems.
For the setting, the second dynamic system was constructed from the first system using finite derivative delay terms that included the boundary points of the considered interval. This study followed the structure of other relevant studies [6] in assuming that the forcing terms of the system are the control forces. This study is organized as follows: Section 2 presents the criteria for the optimal issues. Section 3 presents the approach for determining the minimum energy for the transition procedure. Section 4 presents the numerical results attained by choosing different parameters for various cost requirements. Section 5 presents the summary of this study.

The Model
Consider the class of weakly singular integro-differential equations of the first The difference operator D is defined as The weighting kernel g is integrable, positive, nondecreasing, and weakly singular at 0 = s . The control force ( ) u t is assumed to be locally integrable for 0 > t . Although a more general kernel g also works, this study focused on the Abel-type kernel (i.e., ( ) from the original aeroelastic model).
The initial condition ( ), Note that the initial value problem in Equations (1)-(2) can be written as Dx Dx u (5) provided that the function Dx g s x t s s (6) is absolutely continuous for 0 > t and the function The second system is a class of weakly singular integro-differential equations of the second kind where l is a positive integer and 0 1, 1, , For the partition between systems (2)  λ ∈ is assumed. Therefore, the combined system can be written as Although the proposed methods can be applied to more general cost functions, this study primarily considered the typical cost function for comparison: where h is a constant of final target state, ( ) η t is a target function, and parameters 1 2 , α α and 3 α are nonnegative constants with a total sum of 1.

The Numerical Method
This procedure is proposed to discretize system (9) and the cost function (10) simultaneously to construct two corresponding linear systems with unknowns as states and controls. The space mesh points (corresponding to the s variable) are discretized as , and a new variable ξ is defined System (9) can then be reformulated as a first-order hyperbolic equation with the condition After defining corresponding constants 0 1 , , ,  n c c c , and 0 1 , , ,  n d d d , Equation (23) can be written in the following simplified form: The connection between the solution ( ) x t and α's is as follows: Because can be obtained in the following case: For the cost function Taking the first derivatives of ( )      , , 0,1, 0 α α α = , the problem is the "tracking problem".
Typical cost distribution is as the following two graphs (Figure 1 and Figure 2).

Conclusion
This study presented a numerical method for finding the minimum of the total cost when it contains two partial costs from two dynamic systems, and each cost In other words, dynamic system with the first kind integro-differential equation is the most stable system in the minimum cost sense.