A New Approach for a Class of Optimal Control Problems of Volterra Integral Equations

In this paper, we propose a new approach for a class of optimal control problems governed by Volterra integral equations which is based on linear combination property of intervals. We convert the nonlinear terms in constraints of problem to the corresponding linear terms. Discretization method is also applied to convert the new problems to the discrete-time problem. In addition, some numerical examples are presented to illustrate the effectiveness of the proposed approach.


Bonnens and Vega
In this paper, we are interested to solving the following class of the OCV problem (1)-( 2) which we called it COCV problem: where function is a continuous function.A controlled Volterra integral equation similar to equation ( 4) is discussed in [16].We suppose that where U a compact and connected set.In addition, we let the final state   y T is a given known number.Here, the linear combination property of intervals is used to convert nonlinear controlled Volterra integral Equation (4) to the equivalent linear equation.The new optimal control problem with this linear Volterra integral equation is transformed to a discrete-time problem that could be solved by linear programming methods.This paper organized as follows.Section 2, transforms the nonlinear function to a corresponding function that is linear respect to a new control function.Section 3, converts the new problem to the discrete-time problem via discretization.In Section 4, numerical examples are presented to illustrated effectivness of this approach.
Finally, the conclusion of this paper is given in Section 5.

Metamorphosis of the COCV Problem
In this section, COCV problem (1) is transformed to the new equivalent problem.First, we state and prove the following two theorems: Theorem 2.1: Let be a continuous function on where U is a compact and connected subset of , then for any arbitrary (but fixed)  is a continuous function on U. Since, continuous function preserve compactness and connectedness, the set is compact and connected.Therefore, , we may suppose the lower and upper bounds of interval respectively.Thus we have: In other words Theorem 2.2: Let functions (.) g and be defined by relations ( 6) and (7).Then they are uniformly continuous on .
(.) w [0, ] T Proof: We will show that (.) g is a uniformly continuous function.It is sufficient that we show that for any given 0   , there exists . Since, any continuous function on compact set is a uniformly continuous.The function on compact set is a uniformly continuous, i.e. for any z  .In addition, by ( 5), 1 1 f s u   .Now, by taking infimum on the right hand side of the last inequality   . By a similar procedure, we have g s  .The proof of uniformly continuity of is similar.□ (.) w By linear combination property of intervals and relation (5), we have for any Thus, we transform COCV problem (3)-( 4) by relation ( 6) as the following continuous-time problem: where and Note that in the new problem (9), which is a optimal control of linear Volterra integral equation, (.)    is the new control function.Next section, converts the problem (9) to the corresponding linear programming problem.

Discrete-Time Problem
Now, discretization method enables us transforming continuous problem (9) to the corresponding discrete form.Consider equidistance points 0 where N is a given big number.Also, we set j j t s  for 0,1, , j N   .By trapezoidal approximation in numerical integration, problem ( 9) is converted to the following discrete-time problem:    and   j j q q t  for all .In problem (10)  that is a known number.
By solving problem (10), which is a linear programming problem, we are able to obtain the optimal solution j   and j y  for all 0,1, 2, , j N   .Note that, for evaluating optimal control variable , we must use the following equation:

Numerical Examples
Here, we use our approach to obtain approximate optimal solution of the following two COCV problems by solving linear programming (LP) problem ( 10) via simplex method [26] in MATLAB software.


The optimal solutions j y  and j   , 0,1, j  of problem ( 12) is obtained by solving problem (10) which is illustrated in Figures 1 and 2 respectively.Here, the value of optimal solution of objective function is -0.470.The corresponding Equation (11) for this example is 2, ,100  The optimal control * j u , of this example is showed in  .In this example for all We obtain the optimal solution j y  and j   , 0,1, j  of problem ( 13) by solving problem (10) which is illustrated in Figures 4 and 5 respectively.In 2, ,100 

Conclusions
In this paper, we posed a different approach for a class of nonlinear optimal control problem including Volterra integral equations.In our approach, the linear combination property of intervals is used to obtain the new corresponding problem which is a linear problem.The new problem can be converted to a LP problem by discretezation method.Finally, we obtain an approximate solution for the main problem.In next works, we are going to use our approach for subclasses of problem (1)-( 2) which Volterra integral equation is similar to Equation ( 4), but objective functional is quadratic or nonlinear with respect to state variables.
Consider the following optimal control problem governed by Volterra integral equation (OCV):