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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.

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What is Optimal Control Theory?


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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.
basic concepts and definitions
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.

formulations of simple control models
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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Example 1.2 An Advertising Model.

We consider a special case of the Nerlove-Arrow advertising model.

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Example 1.3 A Consumption Model.

This model is summarized in Table 1.3:

history of optimal control theory
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
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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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
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
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
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  Enis column vector and z  Emis row vector, n 2, m 2, then fz will denote the k x m matrix;

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fy will denote the k x n matrix
  • Matrices fz and fyare 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
product rule for differentiation
Product Rule for Differentiation
  • Let x  Enbe a column vector and g(x)  Enbe a row

vector and f(x)  Enbe a column vector, then

vector norm
Vector Norm
  • The norm of an m-component row or column vector z is defined to be
  • Neighborhood Nzoof a point is

where  > 0 is a small positive real number.

  • A function F(z): Em E1is 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
Some Special Notation
  • left and rightlimits
  • 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.

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bang function

  • sat function
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impulse control

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

convex set and convex hull
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  Enis
concave and convex function
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).

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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 xis a vector, x xneeds 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.
affine function and homogeneous function of degree one
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  En x Em is called a saddle point of (x,y), if
  • Note also that
linear independence and rank of a matrix
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.