Spectral Methods For Some Optimal Control Problems Governed By Differential Equations

Optimal control is one of the most principal components of the control theories. Inpractice, the optimal control problems are widely used in the areas of space?ight, navi-gation and military a?airs, etc. For instance, the optimal low thrust trajectories to themoon [5], the ?ight control of the rocket [27, 68] and the recovery of missiles are suchkind of problems. There are also many kinds of optimal control problems in the indus-trial systems such as the bacterium quantity control in bioengineering, electric powercontrol system and optimal control about sewage disposal of oil field, etc. Thereforeoptimal control gained much attention for its wide applications in the engineering op-timization design and control [58, 84]. In the past two decades, there are noticeabledevelopments of optimal control in control engineering as well as aeronautics control andguide[42, 62, 64, 73].The spectral method is one of the most important numerical methods for solvingdi?erential equations. The main advantage of the spectral method is the so-called"spec-tral accuracy", which is very competitive with its finite di?erence and finite elementcounterparts. The pioneering work of using spectral methods for solving optimal con-trol problems arose in the 1980's [16, 19] and the research of this field has developedfast during the past two decades. Ross and Fahroo applied the pseudospectral (PS)methods to optimal control problems, and attained successful results. They developedChebyshev spectral methods[26], Legendre spectral methods[70, 71] and spectral knottingmethods[74] for solving optimal control problems. In solving the smooth optimal controlproblems, the spectral methods have high accuracy. As for some special constrainedproblems, the controls will distort in certain subdomains, and this kind of problems canbe resolved by the spectral element methods. With the development of spectral meth-ods, especially the appearance of domain decomposition methods and their applicationsto numerical solutions of di?erential equations, some scholars managed to use the corre-sponding methods to solve optimal control problems. Among those, a remarkable job isthe spectral patching method for optimal controls [65]. The advantage of this method isthat the relatively lower order polynomials could be used to approximate the control andthe state functions over each subinterval. In this way, the method could be more ?exibleand we can reallocate the collocation nodes according to di?erent cases.The outline of this dissertation is as follows:We first introduce the Chebyshev–Legendre (CL) method for solving the optimalcontrol problems. Although the CL method has been developed fast in solving di?erentialequations [51, 52, 78], it has not been generalized to solve optimal control problems. Thismethod allows us to use the Legendre polynomials as the basis functions and, at the sametime, employ the Chebyshev–Gauss–Lobatto (CGL) points as the collocation points. Byusing the Chebyshev–Legendre method along with the fast Legendre transform, the CPUcomputing time is reduced e?ciently, which is one of the most significant advantages ofthis method. On the other hand, the employed CGL points have explicit forms, whichlead to easiness of their obtaining. It is for computation simplicity and high accuracy thatwe prefer to use the CGL points as the collocation nodes in our pseudospectral method.In Chapter 3, we mainly discuss about the linear–quadratic optimal control problemsand give an e?ective numerical solution scheme. In Chapter 4, we use the CL method tosolve both the general type OC problems and the problems applied in engineering. Thenumerical experiment results further illustrate the e?ectiveness of the proposed method. We also give the convergence results and the proof of some theorems, which are necessaryto support the method.As for the nonsmooth optimal control problems, we consider using the multidomainmethod, say, piecewise Chebyshev–Legendre method to solve it. That is the work listedin Chapter 5. A series of techniques are employed to reduce the errors in numerical calcu-lation. Good results are performed. We employed higher order interpolation polynomialsat the interspace where the solution is smooth, while we utilized lower order polynomialsin the domain where the solution is not smooth enough. Thus the overall precision isimproved. The convergence theories and numerical experiments illustrate the feasibilityand e?ciency of the method.Moreover, in Chapter 6, we give an improved Chebyshev pseudospectral method tosolve the optimal control problems. We also use the proposed method to solve boththe general and the special problems in engineering. The unknowns of the nonlinearprogramming problems are the Chebyshev expansion coe?cients instead of the functionvalues at collocation points. The fast cosine transform based on FFT is used to reducethe CPU time cost.Furthermore, we study a class of optimal control problems governed by second orderlinear elliptic equation in Chapter 7. We give the CL spectral discrete scheme for solvingthe problem and derive a priori error estimates of the method. Several academic examplesare conducted via using the methods.