Optimal Control And Stability Of Nonlinear Piecewise Smooth Dynamical Systems

This dissertation, based on the engineering background of design of 3D-trajeetory of horizontal wells, studies the optimal control problem of nonlinear piecewise smooth dynamical systems. Furthermore, the qualitative analysis on the stability of piecewise smooth systems with impulses at fixed times are discussed, and the stability analysis of nonlinear dynamical systems with initial time difference are made. These results can not only develop the theory and application of nonlinear dynamical systems and optimal control, but also provide some guidance for the design of the trajectory of horizontal wells. Therefore, this research is very interesting in both theory and practice. The main results, obtained in this dissertation, may be summarized as follows:1. According to the features of 3D-trajectory formed in horizontal wells, we construct a nonlinear piecewise smooth dynamical system in which the state variables are inclination, azimuth, north coordinate, east coordinate and vertical depth coordinate and the controls are radius, tool-face angle and arc length. Taking the weight sum of precision of hitting target and the total length of the trajectory as a performance criterion, we construct an optimal control model of the trajectory of horizontal wells. The necessary conditions for optimality of nonlinear piecewise smooth dynamical system are proved via maximum principle. Because the performance criterion is highly nonlinear and computationally expensive to evaluate, it limits the efficient use of classical gradient based optimization methods. So we construct a new algorithm in which the uniform design technique has been incorporated into the revised Hooke-Jeeves algorithm to handle the multimodal function. Firstly, we use uniform design method to generate many initial points in control domain, and decompose the control domain into many subdomains. Then we get the locally optimal solutions in each subdomains by the revised Hooke-Jeeves algorithm. It is shown from the real example that the revised Hooke-Jeeves method is efficient. Furthermore, we take fully into account the effect of unknown disturbances in drilling. We present an impulsive optimal control model to solve the optimal designing problem of the trajectory of horizontal wells with disturbances, and we prove that the optimal control exists. To obtain the optimal solutions, the optimal control problem can be converted into a nonlinear parametric optimization by integrating the state equation. The general algorithms for nonlinear parametric optimization problems are all need to calculate the gradient of objective function. Since it is difficult to achieve the gradient of our objective function, these algorithms are not appropriate for our nonlinear parametric optimization problem. We discuss here that the locally optimal solution depends in a continuous way on the parameters (disturbances) and utilize this property to propose a revised Hooke-Jeeves algorithm. The numerical simulation is in accordance with theoretical results. The numerical results illustrate the validity of the model and efficiency of the algorithm.2. The stability, practical stability and boundedness of impulsive differential systems are proved by using the method of perturbing Lyapunov functions. It is hard to find a Lyapunov function satisfying all the desired conditions. We obtain the stability, asymptotical stability, practical stability and boundedness properties of nonlinear impulsive differential systems using perturbing Lyapunov functions in a manner that provide weaker assumptions. We generalize the concepts of perturbing Lyapunov functions to impulsive differential systems and prove several stability results of nonlinear impulsive systems in terms of two measures by perturbing Lyapunov functions. We find that the method of perturbing Lyapunov functions yields better results since each function can satisfy less rigid requirement.3. This dissertation investigates the stability and boundedness of nonlinear differential dynamical systems relative to initial time difference. The traditional stability concept assumes that only state variable perturbs and the initial time keeps unchanged. However, it is impossible not to make errors in the starting time in practical applications. Therefore, it is important to study the deviation in initial time. We establishes several criteria on stability and boundedness for nonlinear differential dynamical systems relative to initial time difference by employing a new comparison principle. The conditions for our results are more easy to test in applications than the existing results. We initiate three examples to test our results.