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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.

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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:

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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.

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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.

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Table 1.1 The Production-Inventory Model of Example 1.1

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Example 1.2 An Advertising Model. We consider a special case of the Nerlove-Arrow advertising model.

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Table 1.2 The Advertising Model of Example 1.2

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Example 1.3 A Consumption Model. This model is summarized in Table 1.3:

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Table 1.3 The Consumption Model of Example 1.3

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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

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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.,

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when n = m, we can define the inner product

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If is an m x k matrix and B={b ij } is a k x n matrix, C={c ij }=AB, which is an m x n matrix with components:

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Differentiating Vectors and Matrices with respect to Scalars Let f : E 1 E k be a k-dimensional function of a scalar variable t. If f is a row vector, then If f is a column vector, then

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Differentiating Scalars with respect to Vectors If F: E n x E m E 1, n 2, m 2, then the gradients F y and F z are defined, respectively as;

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Differentiating Vectors with respect to Vectors If F: E n x E m E k, is a k – dimensional vector function, f either row or column, k 2; i,e; where f i = f i (y,z), y E n is column vector and z E m is row vector, n 2, m 2, then f z will denote the k x m matrix;

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f y will denote the k x n matrix Matrices f z and f y are known as Jacobian matrices. Applying the rule (1.11) to F y in (1.9) and the rule (1.12) to F z in (1.10), respectively, we obtain F yz =(F y ) z to be the n x m matrix

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and F zy = (F z ) y to be the m x n matrix

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Product Rule for Differentiation Let x E n be a column vector and g(x) E n be a row vector and f(x) E n be a column vector, then

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Vector Norm The norm of an m-component row or column vector z is defined to be Neighborhood N zo of a point is where > 0 is a small positive real number. A function F(z): E m E 1 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

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Some Special Notation left and right limits x k : state variable at time k. u k : control variable at time k. k : adjoint variable at time k, k=0,1,2,…,T. : difference operator. x k := x k+1 - x k. x k *, u k *, and k, are quantities along an optimal path. discrete time (employed in Chapters 8-9)

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sat function bang function

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impulse control If the impulse is applied at time t, then we calculate the objective function J as

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Convex Set and Convex Hull A set D E n is a convex set if y, z D, py +(1-p)z D, for each p [0,1]. Given x i E n, i=1,2,…,l, we define y E n to be a convex combination of x i E n, if p i 0 such that The convex hull of a set D E n is

26 Concave and Convex Function : D E 1 defined on a convex set D E n 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

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If the function is twice differentiable, then it is concave if at each point in [a,b], x x 0. In case x is a vector, x x needs to be a negative definite matrix. If : D E 1 defined on a convex set D E n is a concave function, then - : D E 1 is a convex function.

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Figure 1.2 A Concave Function

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Affine Function and Homogeneous Function of Degree One : E n E 1 is said to be affine, if (x) - (0) is linear. : E n E 1 is said to be homogeneous of degree one, if (bx) = b (x), where b is a scalar constant. Saddle Point : E n x E m E 1, a point E n x E m is called a saddle point of (x,y), if Note also that

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Figure 1.3 An Illustration of a Saddle Point

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Linear Independence and Rank of a Matrix A set of vectors a 1,a 2,…,a n E n is said to be linearly dependent if p i,, not all zero, such that If the only set of p i for which (1.25) holds is p 1 = p 2 = ….= p n = 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.

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