Minimum Principle of Pontryagin

Minimum Principle of Pontryagin.

TABLE CONTENTS

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1 Linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.1 A basic result concerning linear maps . . . . . . . . . . . 8
1.1.2 Bounded Linear Maps . . . . . . . . . . . . . . . . . . . . 9
1.2 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Dierential Calculus in Banach spaces . . . . . . . . . . . . . . . 13
1.5 Convex sets and convex functions . . . . . . . . . . . . . . . . . . 14
1.5.1 Notation and Further denitions . . . . . . . . . . . . . . 17
1.6 Lower Semi-Continuous Functions . . . . . . . . . . . . . . . . . 18
1.7 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.8 Optimality condition: . . . . . . . . . . . . . . . . . . . . . . . . 21
1.9 Optimization with equality constrains . . . . . . . . . . . . . . . 23
2 Pontryagin minimum method principle 26
2.0.1 Towards the principle of pontryagin . . . . . . . . . . . . 28
3 Minimum Principle of Pontryagin: Linear Quadratic Case 30
3.1 Existence and uniqueness of the optimal control . . . . . . . . . . 31
3.2 characterization of the optimal control . . . . . . . . . . . . . . . 35
3.2.1 Riccati equation . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

PRELIMINARIES

Linear maps  In this part we define linear maps and present some basic results concerning them. Definition: Let X and linear spaces over a scalar eld K.A mapping T: X −→ Y is said to be a linear map if T (αx + βy) = αT(x) + βT(y) (1.1) for arbitrary x, y ∈ X and arbitrary scalars α, β ∈ K. Some authors use the term linear operator or linear transformation instead of a linear map. Condition (1.1) is equivalent to the following two conditions: (i)T(x + y) = T(x) + T(y)∀x, y ∈ X (ii)T(αx) = αT(x)∀x ∈ X and for each scalar, α.  

A basic result concerning linear maps We remark rst that since linear functionals are special forms of linear maps, any result proved for linear map holds for linear functionals. Proposition 1.1.1 Let X and Y be two linear spaces over a scalar eld, K, and let T: X −→ Y be a linear map. Then 1. T (0) = 0 2. The rang of T, R(T) = {y ∈ Y: T(x) = y for some x ∈ X} is a linear subspace of Y 3. T is one to one if and only if T(x) = 0 implies that x = 0 4. If T is one to one, then T −1 exists on R(T) and T −1: R(T) −→ X is also a linear map.  

Proof. (1) Since T is linear, we have, T(αx) = αT(x) for each x ∈ X and each scalar α. Take α = 0 and (1) follows immediately. (2) We need to show that for y1, y2 ∈ R(T) and α, β scalars, αy1 + βy2 ∈ R(T). Now, y1, y2 ∈ R(T) implies that there exists x1, x2 ∈ X such that T(x1) = y1, T(x2) = y2. Moreover, αx1 + βx2 ∈ X (since X is a linear space). Furthermore, by the linearity of T, we have T (αx1 + βx2) = αT(x1) + βT(x2) Hence αy1 + βy2 ∈ R(T), and so R(T) is a linear subspace of Y. (3) (⇒) Assume that T is one to one. Clearly T(x) = 0 ⇒ T(x) = T (0) since T is linear (and so T (0) = 0).  

But T is one-to-one. So, x = 0. (⇐) Assume that whenever T(u) = 0 then u must be 0. We want to prove that T is one-to-one. So, let T(x) = T(y). Then T(x) − T(y) = 0 and by the linearity of T, T (x − y) = 0. By hypothesis, x − y = 0 which implies x = y. Hence T is one to-one. (4) Assume that T is one-to-one since the restriction of T on R(T) is always onto, then T is bijective from X into R(T). So, T −1 exists on R(T). For the linearity, let y1, y2 ∈ R(T) and α a scalar. 

Then there exists x1, x2 ∈ X such that y1 = T(x1), y2 = T(x2). so T −1 (y1 + αy2) = T −1 (T(x1) + αT(x2) which is equivalent to: T −1 (y1 + αy2) = T −1 (T(x1 + αx2)) by using the linearity of T, which is also equivalent to T −1 (y1 + αy2) = x1 + αx2 = T −1 (y1) + αT −1 (y2) Therefore T −1 is linear.

BIBLIOGRAPHY

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N. Djitte ; Cours Optimisation en dimension innie: Calcul des Variations,Controle Optimal: Principe de Pontryagin, Universite Gaston Berger,Saint
Louis, Senegal 2005-2006.
N. Djitte, Lecture notes on Convex analysis and Variational method,
African university of science and Technology, 2011
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