Optimization Methods For Finding Periodic Solutions Of Differential Equations

In this paper, we are mainly concerned with finding periodic solutions of differential equations using optimization methods and solving the nonlinear programming problem using the interior-point quasi-Newton method.The periodic solution of differential equations is a very active research topic in the current mathematical physics and applied mathematics field. It can express not only some periodic motions but also some aperiodic motions. The problem of finding periodic solutions of differential equations has attracted great attentions of many mathematicians , such as Poincare, Levinson, Yoshizawa, Massera, Burton etc. Until now, plenty of results on the periodic solutions of differential equations have been obtained by using some kinds of methods and techniques, seen the references[2, 60,11,20, 29, 30, 37, 39, 40, 43, 44, 52, 58, 67, 68].â… .In chapter 2, Firstly, we consider the Duffing equation:where f(t, x) is continuous and periodic in t with period T > 0 and continuously differentiable in x. Theorem 1 If the equation(1) satisfies the fundamental nonresonant conditionthen equation(1) has a unique T-periodic solution.Secondly, we consider the Li(?)nard-type equation:where f(t,x) is continuous and periodic in t with period T > 0 and continuously differentiable in x,a(t) and g(t) are T-periodic continuous functions.Theorem 2 Assume∫₀^T a(t)dt = 0 and f(t,x) satisfies the following nonresonant conditionwhere constantsα,βsatisfyfor some nonnegative integer N. Then the Lienard-type equation (3) has a unique T-periodic solution.Let x(t,P) be the solution of the equation (1) and (3) with the initial valuesAccording to the Poincare map method, the equation (1) and (3) have a T-periodic solution, if and only if there exists a point P^*=(P₀^*,P₁^*)^T∈R² such that Thus, for the following optimization problemwhere X(T, P)=(x(T, P),x'(T, P))^T,X(0,P)=(x(0,P),x'(0, P))^T=P, if there exists a global optimal solution P^* such that J(P^*) = 0, then the solution of equation (1) and (3) starting from P^* is the periodic solution.By the optimization theory, the KKT condition for the unconstrained optimization problem (8) isTheorem 3 Consider the Duffing equation (1) satifying the condition (2) and the Li(?)nard-type equation (3) satisfying the condition (4)-(5). If there exist P^k∈R²(k =1,2,…) such thatThenandThe theorem 3 shows that the problem of finding the periodic solutions of the equation (1) and (3) can be converted into the one of finding the global optimal solutions of the unconstrained optimization (8). The quasi-Newton method is an effective and famous method to solve the unconstrained optimization problem. It substitutes the matrix without the second-derivative for the Hessian or the inverse Hessian matrix in the Newton method, and holds high convergence rate. Therefore, we adopt the quasi-Newton method to solve the problem (8).The quasi-Newton equation:where H^k+1 is the approximation of the inverse Hessian (?)²J(P^k)^-1 of the objective function J(P^k),η^k=P^k+1-P^k,γ^k=g^k-1-g^k.We adopt the DFP algorithm:â…¡.In Chapter 3, we study the modified augmented Lagrangian line-search interior-point quasi-Newton method for nonlinear programming.In 2002, Argaez and Tapia construct an appropriate merit function that couples the objective function with the constraints in such a way that progress in the merit function effectively means real progress in solving the optimization problem. The choice of merit function was influenced by the merit function suggested by [12] for their convex programming application. The strategy is to modify the augmented Lagrangian function associated with the equality constrained optimization problem by adding to its penalty term a potential reduction function utilized in linear programming to handle the perturbed complementarity conditions. Consider the general nonlinear program in the form:where f:Rⁿ→R and h :Rⁿ→R^m are twice continuously differential functions.The Lagrangian function associated with problem (14) is:where y∈R^m and z≥0∈Rⁿ are Lagrange multipliers associated with the equality and inequality constraints, respectively.The set of indices of active or binding inequality constraints isl(x)= {j :x_j = 0}.In this chapter, We make the following assumptions:(A1) ( Existence ) There exists x^* a solution to problem (19).(A2) ( Smoothness ) The Hessian operators (?)²f,(?)²h_i,i=1,…m,are Lipschitz continuous in a neighborhood of x^*.(A3) ( Regularity ) The set {(?)h₁(x^*),…,(?)h_m(x^*)}∪{e_j :j∈I(x^*)} is linearly independent.(A4) ( Strict Complementarity ) For all j,x_j^* + z_j^* > 0.Forμ>0,the perturbed KKT conditions associated with problem (14) are given by :where e = (1,...,1)^T∈Rⁿ,X = diag(x),Z = diag(z). Forμ= 0, the condiction (16) is equivalent to the KKT conditions of the problem (14).Definition 1 For anyμ> 0,define the generalized augmented Lagrangian function by:where M_μ:R^n+m+n→R,ι(x,y, z) is the Lagrangiang function, p≥0 is a penalty parameter, andΦ_μ(x,z) is the penalty term:By construction of the Merit function M_μ,the primary variables x, z will be kept positive.Definition 2 An interior point (x, y, z) is said to be a quasicentral point for a givenμ>0 ifThe quasicentral path associated with problem (14) is defined as the collection of quasicentral points (18) and is parameterized byμ.Observe that a point x is the first component of a point on the quasicentral path, i.e., if and only if x is strictly feasible, i.e.,x∈S={x∈Rⁿ:h(x)=0,x>0}.Definition 3 We defineα(μ,γ-neighborhood of a point on the quasicentral path corresponding toμby:where (μ,γ)>0,w=(XZe)^1/2.We call the valueγμthe radius of this neighborhood.The definition of N_μ(γ) give us a measure of how close an interiorpoint is satisfying the perturbed KKT conditions for a correspondingμ>0. The basic ideea of our global algorithm is to apply a linesearch quasi-Newton method to the perturbed condition for fixedμuntil we arrive to a specified (μ,γ)- neighborhood. Then we decreaseμ,specify a new (μ,γ)-neighborhood, and repeat the linesearch quasi-Newton method.