The aim of this article is to present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. It is based on recent theoretical developments of geometric optimal control, and the article contains a review of second order optimality conditions. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity of the extremal flow. We derive an algorithm called COTCOT (Conditions...

In this article, a new solution to the steering control problem of nonholonomic systems, which are transformable into chained form is investigated. A smooth super twisting sliding mode control technique is used to steer nonholonomic systems. Firstly, the nonholonomic system is transformed into a chained form system, which is further decomposed into two subsystems. Secondly, the second subsystem is steered to the origin by using smooth super twisting sliding mode control. Finally, the first subsystem...

A numerical technique for solving the classical brachistochrone problem in the calculus of variations is presented. The brachistochrone problem is first formulated as a nonlinear optimal control problem. Application of this method results in the transformation of differential and integral expressions into some algebraic equations to which Newton-type methods can be applied. The method is general, and yields accurate results.

In some preceding works we consider a class $\mathcal{O}\mathcal{P}$ of Boltz optimization problems for Lagrangian mechanical systems, where it is relevant a line $l=l_{\gamma (\cdot )}$, regarded as determined by its (variable) curvature function $\gamma (\cdot )$ of domain $\left[s_0,s_1\right]$. Assume that the problem $\tilde{\mathcal{P}}\in \mathcal{O}\mathcal{P}$ is regular but has an impulsive monotone character in the sense that near each of some points ${\delta }_1$ to $\delta {}_{\nu }\gamma (\cdot )$ is monotone and $\left|\gamma {}^{\prime }\right(\cdot \left)\right|$ is very large. In [10] we propose a procedure belonging to the theory of impulsive controls, in order to simplify $\tilde{\mathcal{P}}$ into a structurally...

Synchronization with error bound of two non-identical forced oscillators is studied in the paper. By introducing two auxiliary autonomous systems, differential inequality technique and active control technique are used to deal with the synchronization of two non-identical forced oscillators with parameter mismatch in external harmonic excitations. Numerical simulations show the effectiveness of the proposed method.