In this paper, a method for the indirect solution of the optimal control problem (OCP) in the presence of pure state variable inequality constraints (SVICs) and mixed state-control inequality constraints (SCIC), without a need for a close initial guess is presented. In the proposed method, using the finite difference approximation (FDA), the pure SVICs are converted to SCIC. Here, the distance of the constraint function to the feasibility bounds of the constraint is computed in every situation and the control signal is chosen appropriately to facilitate the constraint stays safe. In this method, prior knowledge of the numbers and sequences of activation times is not required. So, it can be simply implemented in continuous boundary value problem (BVP) solvers. The proposed method simply applies the SVICs and since the constraint is directly applied on the control signal, it improves the convergence. On the other hand, because of the convergence problem in the indirect solution of OCP, the simple homotopy continuation method (HCM) is used to overcome the initial guess problem by deploying a secondary OCP for which the initial guess can be zero. The proposed approach is applied on a few comprehensive problems in the presence of different constraints. Simulations are compared with the direct solution of the OCP to confirm the accuracy and with the penalty function method and the sequential constraint-free OCP to confirm the convergence. The results indicate that the FDA method for handling the constraints along with the HCM is easy to apply with acceptable accuracy and convergence, even for highly nonlinear problems in robotic systems such as the constrained time optimal control of a two-link manipulator (TLM) and a three-link common industrial robot.

For the third-order differential equation y′″ = ƒ(t, y, y′, y″), where , questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

The Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease ina population of constant size is considered. In order to control the spread of infection,we propose the model with four bounded controls which describe vaccination of newborns,vaccination of the susceptible, treatment of infected, and indirect strategies aimed at areduction of the incidence rate (e. g. quarantine). The optimal control problem ofminimizing the total number of the infected individuals on a given time interval is statedand solved. The optimal solutions are obtained with the use of the Pontryagin MaximumPrinciple and investigated analytically. Numerical results are presented to illustrate theoptimal solutions.

The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained.

We consider an optimal control problem describing a laser-inducedpopulation transfer on a n-level quantum system. For a convex cost depending only on the moduliof controls (i.e. the lasers intensities),we prove that there always exists a minimizer inresonance. This permits to justifysome strategies used in experimental physics. It is also quite importantbecause it permits to reduce remarkablythe complexity of the problem (and extend some of our previous resultsfor n=2 and n=3): instead of looking for minimizers on thesphere $S^{2n-1}\subset\mathbb{C}^n$ one is reduced to look just forminimizers on the sphere $S^{n-1}\subset \mathbb{R}^n$ . Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.

In this paper we consider the well-known implicit Lagrange problem: find a trajectory solution of an underdetermined implicit differentialequation, satisfying some boundary conditions and which is a minimum of the integral of a Lagrangian. In the tangent bundle of the surroundingmanifold X, we define the geometric framework of q-pi- submanifold. This is an extension of the geometric framework of pi- submanifold,defined by Rabier and Rheinboldt for determined implicit differential equations, to underdetermined implicit differential equations. With thisgeometric framework we define a class of well-posed implicit differential equations for which we locally obtain, by means of a reductionprocedure, a controlled vector field on a submanifold W of the surrounding manifold X. We then show that the implicit Lagrange problem leadsto, locally, an explicit optimal control problem on the submanifold W for which the Pontryagin maximum principle is naturally used.

The problem of scheduling items for service with random service times is formulated as an optimal control problem. Pontryagin's maximum principle is used to determine the optimal schedule in certain cases.