Presenting two new methods for solving linear interval optimal control problems using the resilience approach of control signal

In this paper, two new methods are presented to find the optimal control of systems described by linear differential equations that have interval coefficients. In these methods, the given interval optimal control problem is transformed into a non-interval (deterministic) optimal control problem so that it is possible to use the optimal control theory to solve it. By considering the variation limits for the control signal, the optimal solution of the main problem is obtained.

In this research, two methods are presented to solve the deterministic sub-models of linear interval optimal control problems and find a good approximation for the optimal solution of them. In the first method, by using the center of intervals, the given interval optimal control problem becomes a deterministic optimal control problem and it is possible to use the Pontriagin conditions to solve it. By considering a resilience variable for the resulting solution, the limits of the control signal changes and the interval optimal solution for the main problem are obtained. In the second method, we first solve the optimal control problem for the beginning and the end of the interval, and at any time we consider the lowest value as the lower bound and the highest value as the upper bound of the interval solution. Then, by applying the constraints of the problem, we calculate the resilience of the state variable and the control signal. We also used Hokuhara's difference to reduce the error in interval calculations.

Solving optimization problems using the interval optimal control approach, show that the optimal solution of the submodel that is solved by any of these two methods is within the range of the solution of the main problem. In addition, the solutions of these problems are also presented as interval values.

Originality/Value:

 To obtain the numerical results, MATLAB software and INTLAB software package of interval calculations have been used. These approaches are categorized into indirect methods for solving optimal control problems, while it is far from their defects, e.g. curse of burdensome computational load, so that interval approach is applied to simply solve the problems.