What is Optimal Control Theory?   Dynamic Systems: Evolving over time. Time: Discrete or continuous. Optimal way to control a dynamic system.  Prerequisites: Calculus, Vectors and Matrices, ODE&PDE  Applications: Production, Finance/Economics, Marketing and others. Basic Concepts and Definitions  A dynamic system is described by state equation: where x(t) is state variable, u(t) is control variable.  The control aim is to maximize the objective function: Usually the control variable u(t) will be constrained as follows:   Sometimes, we consider the following constraints: (1) Inequality constraint (2) Constraints involving only state variables (3) Terminal state where X(T) is reachable set of the state variables at time T. Formulations of Simple Control Models  Example 1.1 A Production-Inventory Model. We consider the production and inventory storage of a given good in order to meet an exogenous demand at minimum cost. Table 1.1 The Production-Inventory Model of Example 1.1  Example 1.2 An Advertising Model. We consider a special case of the Nerlove-Arrow advertising model. Table 1.2 The Advertising Model of Example 1.2  Example 1.3 A Consumption Model. This model is summarized in Table 1.3: Table 1.3 The Consumption Model of Example 1.3 History of Optimal Control Theory     Calculus of Variations. Brachistochrone problem: path of least time Newton, Leibniz, Bernoulli brothers, Jacobi, Bolza. Pontryagin et al.(1958): Maximum Principle. Figure 1.1 The Brachistochrone problem Notation and Concepts Used       = “is equal to” or “is defined to be equal to” or “is identically equal to.” := “is defined to be equal to.”  “is identically equal to.”  “is approximately equal to.”  “implies.”  “is a member of.” Let y be an n-component column vector and z be an m-component row vector, i.e., when n = m, we can define the inner product If is an m x k matrix and B={bij} is a k x n matrix, C={cij}=AB, which is an m x n matrix with components: Differentiating Vectors and Matrices with respect to Scalars   Let f : E1Ek be a k-dimensional function of a scalar variable t. If f is a row vector, then If f is a column vector, then Differentiating Scalars with respect to Vectors  If F: En x Em  E1, n 2, m 2, then the gradients Fy and Fz are defined, respectively as; Differentiating Vectors with respect to Vectors  If F: En x Em  Ek, is a k – dimensional vector function, f either row or column, k 2; i,e; where fi = fi (y,z), y  En is column vector and z  Em is row vector, n 2, m 2, then fz will denote the k x m matrix; fy will denote the k x n matrix  Matrices fz and fy are known as Jacobian matrices.  Applying the rule (1.11) to Fy in (1.9) and the rule (1.12) to Fz in (1.10), respectively, we obtain Fyz=(Fy)z to be the n x m matrix and Fzy = (Fz)y to be the m x n matrix Product Rule for Differentiation  Let x  En be a column vector and g(x)  En be a row vector and f(x)  En be a column vector, then Vector Norm  The norm of an m-component row or column vector z is defined to be  Neighborhood Nzo of a point is where  > 0 is a small positive real number.  A function F(z): Em  E1 is said to be of the order o(z), if  The norm of an m-dimensional row or column vector function z(t), t  [0, T], is defined to be Some Special Notation   left and right limits discrete time (employed in Chapters 8-9) xk: state variable at time k. uk: control variable at time k. k : adjoint variable at time k, k=0,1,2,…,T. xk:= xk+1- xk.: difference operator. xk*, uk*, and k, are quantities along an optimal path.   bang function sat function impulse control If the impulse is applied at time t, then we calculate the objective function J as Convex Set and Convex Hull    A set D  En is a convex set if  y, z  D, py +(1-p)z  D, for each p  [0,1]. Given xi  En, i=1,2,…,l, we define y  En to be a convex combination of xi  En, if  pi  0 such that The convex hull of a set D  En is Concave and Convex Function     : D  E1 defined on a convex set D  En is concave if for each pair y,z  D and for all p  [0,1], If  is changed to > for all y,z  D with y  z, and 0<p<1, then  is called a strictly concave function. If (x) is a differentiable function on the interval [a,b], then it is concave, if for each pair y,z [a,b], (z)  (y)+  x (y)(z-y).   If the function  is twice differentiable, then it is concave if at each point in [a,b],  xx 0 . In case x is a vector,  x x needs to be a negative definite matrix. If  : D  E1 defined on a convex set D  En is a concave function, then -  : D  E1 is a convex function. Figure 1.2 A Concave Function Affine Function and Homogeneous Function of Degree One   : En  E1 is said to be affine, if (x) - (0) is linear.  : En  E1 is said to be homogeneous of degree one, if (bx) = b (x), where b is a scalar constant. Saddle Point    : En x Em  E1, a point point of (x,y), if Note also that  En x Em is called a saddle Figure 1.3 An Illustration of a Saddle Point Linear Independence and Rank of a Matrix     A set of vectors a1,a2,…,an  En is said to be linearly dependent if  pi,, not all zero, such that If the only set of pi for which (1.25) holds is p1= p2= ….= pn= 0, then the vectors are said to be linearly independent. The rank of an m x n matrix A is the maximum number of linearly independent columns in A, written as rank (A). An m x n matrix is of full rank if rank (A) = n.