mehdi allahdadi

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.

The main aim of this paper is to find an optimal interval control law for quadratic linear problems under interval uncertainty. For this purpose, using Bellman's optimality principle and interval Hamilton-Jacobi-Bellman inequalities, the interval optimal control problem is transformed into a system of interval differential inequalities. These inequalities are called Riccati's differential inequalities, which is a result of the dynamic programming method. To solve this system of inequalities, we use inclusion relations and interval arithmetic. By this method, we can obtain the upper and lower bounds of the solutions. We also use Hokuhara's generalized difference to reduce errors in the interval arithmetic. We apply the presented method for solving some interval quadratic linear optimal control problems by using MATLAB software. The obtained results show the efficiency of the proposed method.

Determining efficient solutions of the Interval Multi Objective Linear Fractional Programming (IMOLFP) model is generally an NP-hard problem. For determining the efficient solutions, an effective method has not yet been proposed. So, we need to have an appropriate method to determine the efficient solutions of the IMOLFP. For the first time, we want to introduce algorithms in which the strongly and weakly efficient solutions of the IMOLFP are obtained.

In this paper, we introduce two algorithms such that in one, strongly feasible of inequalities and in the other, weakly feasible of inequalities are considered (A system of inequalities is strongly feasible if and only if the smallest region is feasible, and a system of inequalities is weakly feasible if and only if the largest region is feasible). We transform the objective functions of the IMOLFP to real linear functions and t‎hen convert to a single objective linear model and then in each iteration of the algorithm, we add some new constraints to the feasible region. By selecting an arbitrary point of the feasible region as start point and using the proposed algorithms, we obtain the strongly and weakly efficient solutions of the IMOLFP.

 In both proposed algorithms, we obtain an efficient solution by selecting the arbitrary points, and by changing the starting point, we obtain a new point as the efficient solution.

In this research, for the first time, we have been able to obtain the strongly and weakly efficient solutions of the IMOLFP.

Uncertainty exists in many real-life engineering and mechanical problems. Here, we assume that uncertainties are caused by intervals of real numbers. In this paper, we consider the interval nonlinear programming (INLP) problems where the objective function and constraints include interval coefficients. So that the variables are deterministic and sign-restricted. Additionally, the constraints are considered in the form of inequalities. A basic task in INLP is calculating the optimal values range of objective function, which may be computationally very expensive. However, if the boundary functions are available, the problems become much easier to solve. By making these assumptions, an efficient method is proposed to compute the optimal values range using two classic nonlinear problems. Then, the optimal values range are obtained by direct inspection for a special kind of interval polynomial programming (IPP) problems. Two numerical examples are given to verify the effectiveness of the proposed method.

There are several methods to compute the optimal bounds of the objective function for interval quadratic programming (IQP) problems, but no method has yet been suggested to calculate a set of optimal solutions of IQP problems. This paper presents an accurate set of optimal solutions for the interval quadratic programming problems. The optimal solution of the quadratic programming problem is not essentially an extreme point. We rst propose conditions that make the optimal solutions of the IQP to extreme points and then, using these conditions, we compute the exact set of optimal solutions for the IQP problem.

In recent decades zirconium oxide has been introduced in the field of dentistry as a high-strength ceramic. Unlike its mechanical advantages, however, due to its inert chemical properties, it bonds poorly to other substrates, so improving bonding strength to an adhesive material is necessary.

In this experimental study, 70 ceramic zirconia blocks were prepared and distributed randomly among 7 groups. Then the shear bond strengths were determined and the samples were examined by a scanning electron microscope (SEM). Statistical analysis was performed by one-way ANOVA and multiple Tukey comparisons.

One-way analysis of variance (ANOVA) showed that laser irradiation distance has a significant effect on orthodontics brackets bond strength to zirconia-ceramics. Based on the Tukey post hoc test, each group was compared with other groups and the contact mode and 2 mm distance groups showed significantly higher bond strength than other groups (P-value <0.05).

Orthodontic bracket bond strength to zirconia-ceramics will be reduced by increasing Er: YAG laser irradiation distance from samples. The highest bond strength will be achieved when the laser irradiation distance is 2 mm or when the laser beam is in contact with samples.

In this paper, solution space of interval linear programming (ILP) models that is a NP-hard problem, has been considered. In all of the solving methods of the ILP, feasibility condition has been only considered. Best-worst case (BWC) is one of the methods for solving the ILP models. Some of the solutions obtained by the BWC may result in an infeasible space. To guarantee that solution is completely feasible, improved two-step method (ITSM) is proposed. By using a new approach, we introduce a space for solving ILP models in which by two tests, feasibility and optimality of the obtained space has been guaranteed.