Existence and uniqueness of optimal cyclic discount Tatyana - - PowerPoint PPT Presentation

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Existence and uniqueness of optimal cyclic process with discount

Tatyana Shutkina

Vladimir State University

shutkina@vlsu.ru Nonlinear control and singularities October 24-28, 2010

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Contents

1 Introduction 2 Existence theorem 3 Necessary optimality condition 4 Analysis of switching function 5 Uniqueness theorem

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Definitions

A cycle process is modeled by a smooth control system on the circle with positive velocities only and a control parameter belonging to a smooth closed manifold or a disjoint union of ones with at least two different points. An admissible motion is defined as an absolutely continuous map x from a time interval to the circle such that at each moment of its differentiability the velocity ˙ x belongs to the convex hull of the admissible velocities of the system. A cycle with a period T > 0 is defined as a periodic admissible motion x, x(t + T) ≡ x(t).

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Arnold’s model

In a presence of a continuous profit density f on the circle an

• ptimization of periodic motion could lead to the problem of the

selection of cyclic process with the maximum time averaged profit: 1 T T f(x(t))dt → max . V.I.Arnold shows that in a generic case the optimal strategy exists, uniqueness and is rather simple.

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Problem reformulation

For an admissible motion x we introduce density ρ, ρ(x(t)) = 1/ ˙ x(t). So almost everywhere we have dx(t) = ˙ x(t)dt

• r, taking into account the positiveness of admissible velocities,

dt = ρ(x(t))dx(t). Thus our extremal problem could be rewritten in the form Aρ(f) :=

2π

• f(x)ρ(x)dx/

2π

• ρ(x)dx → max

In such a formulation we need to find a measurable density ρ on the circle which satisfies the constraint r1 ≤ ρ ≤ r2 (1)

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Models with discounts

Analogous results were proved for the case with a positive discount σ > 0 and discount β which can be positive or negative. Aρ(f) :=

2π

• e

−σ

x

• ρ(z)dz

f(x)ρ(x)dx/

2π

• ρ(x)dx → max
• r

Aρ(f) := (−σ)−1

2π

• f(x)d(e−σφ(x))/

2π

• d(φ(x)) → max .

(2) where φ(x) =

x

• ρ(z)dz

Aρ(f) :=

2π

• e

−σ

x

• ρ(z)dz

f(x)ρ(x)dx/

2π

• e

−β

x

• ρ(z)dz

ρ(x)dx → max

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Existence theorem

Theorem For a continuous profit density f and continuous positive constraint functions r1, r2 there exists a measurable density ρmax which satisfies constraint r1 ≤ ρ ≤ r2 and provides exact upper bound of values Aρ(f) over all measurable functions ρ complying with this constraint.

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Necessary optimality condition

Theorem If for a continuous profit density f and continuous positive constraint functions r1, r2 the maximum A of functional in (2) is provided by a density ρ satisfying constraint (1) then at any point x, where ρ is derivative of its integral, the value e

−σ

x

• ρ(z)dz

f(x) − σ

2π

• x

e

−σ

y

• ρ(z)dz

f(y)ρ(y)dy − A (3) is either non-positive or non-negative, or else zero if the value ρ(x) is equal to either r1(x) or r2(x), or else belongs to (r1(x), r2(x)), respectively.

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Switching function

Expression (3) defines a function S, S = S(x). In some sense this function plays the role of switching function. It could be rewritten as S(x) = e

−σ

x

• ρ(z)dz

f(x)+σ

x

• e

−σ

y

• ρ(z)dz

f(y)ρ(y)dy−σP −A, (4) where P =

2π

• e

−σ

y

• ρ(z)dz

f(y)ρ(y)dy is the profit along the cycle. Let c = −σP − A than S(x) = e

−σ

x

• ρ(z)dz

f(x) + σ

x

• e

−σ

y

• ρ(z)dz

f(y)ρ(y)dy + c

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Uniqueness theorem

Theorem (Uniqueness theorem) For differentiable positive profit density f with a finite number of critical points and continuous positive constraint functions r1, r2, which coincide at isolated points only, in the presence of discount the cyclic process with maximum time averaged profit is uniquely defined. Let m, M are the maximum and minimum values of switching function on zero, which means if c < m or c > M than the motion in the cycle c uses only minimum r1 or maximum r2 density,

• respectively. This values exists and for c m (or c M)period of

level of cycle and profit for this level of cycle are constant. It means that time average profit is constant too. The cycle for

• ptimal average profit is cycle for some level c ∈ [m, M].

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

The monotonicity of cycle

Proposition For continuous positive profit density f and continuous positive constraint functions r1, r2, which coincide at isolated points only, period of level of cycle c is continuous increase function on segment [m, M] and has derivative T ′(c) =

• {xi}

(r2(xi) − r1(xi)) e−σϕ(xi)|f ′(xi)| , (5) without values of level, where switching function has zero on critical point or the ends 0, 2π for zero level. xi is switching point.

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

The monotonicity of cycle

For this formulation, it is obviously, than period is continuous. Let there be only one switching point x1, which is not end’s of

• cycle. Switching function varies continuously with c. For

switching functional than for small variations we doesn’t have

• ther switching points. Consequently, density ρ varies on interval

(x1, x1 + ∆x) △x = −△c S′(x) + · · · = −△c e−σϕ(x1)f ′(x1) + . . .

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

The monotonicity of cycle

△T has the same sign as △c, because it equal

r2(x1)−r1(x1) S′(x1)

△c + . . .

• r

r1(x1)−r2(x1) S′(x1)

△c + . . . , when S′(x1) is positive and negative, respectively. △T = △c r2(x1) − r1(x1) e−σϕ(x1)|f ′(x1)| + . . . Dividing last equality by △c and taken the limit by △c → 0, we get dT dc (c) = r2(x1) − r1(x1) e−σϕ(x1)|f ′(x1)| This derivative is positive, if r1(x1) < r2(x1). If there are few switching points than derivative has the same form, namely, it is just the sum of such terms.

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Differentiability of average profit

Let τm and τM be minimum and maximum period of all level cycles, than c : T → c(T) is continuous increasing function on T ∈ [τm, τM]. So A = A(c(T)), T ∈ [τm, τM]. Proposition If number of critical points of differentiable profit density and points where r1(x1) = r2(x1) are finite, then time average profit, which is function on period of cycle level , is differentiable on [τm, τM] and has derivative A′

T (c(T)) = −c(T) + σP(c(T)) + A(c(T))

T . (6)

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Differentiability of average profit

△P is △c

• xi

(r2(xi) − r1(xi))[e−σφ(xi)f(xi) + σ

xi

• e−σφ(y)f(y)ρ(y)dy − σP(c)]

e−σϕ(xi)|f ′(xi)| we use fact that S(xi) = 0 △P = −△c[c + σP(c)]

• xi

(r2(xi) − r1(xi)) e−σϕ(xi)|f ′(xi)| + . . .

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Differentiability of average profit

We use formula (5) and find PT (c) = −c − σP(c) A′

T (c) = −c + σP(c) + A(c)

T . (7) Consequently, that A′

T (c(T)) = −c(T) + σP(c(T)) + A(c(T))

T .

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

Proof of uniqueness theorem

If for τ ∈ [τm, τM], A′

T (c) = 0, we get

c(τ) + σP(c(τ)) + A(c(τ)) = 0

• r

− c(τ) − σP(c(τ)) = A(c(τ)), The second derivative is A′′

T (c(T)) = −cT (T) − σ(c(T) + σP(c(T))) + AT (c(T))

T + +c + σP(c) + A(c) T 2 . So at the point τ A′′

T (c(τ)) = −cT (τ) + σA(c(τ))

T ,

Existence and uniqueness

• f optimal

cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary

• ptimality

condition Analysis of switching function Uniqueness theorem

References

V.I. Arnol’d Averaged optimization and phase transition in control dynamical systems// Funct. Anal. and its Appl. 36 (2002), 1-11. A.A. Davydov Generic profit singularities in Arnold’s model of cyclic processes// Proceedings of the Steklov Institute of mathematics, V.250 , 70-84, (2005). Davydov A., Shutkina T. Time averaged optimization of cyclic processes with discount. Nonlinear Analysis and optimization problems. Proceedings from the International conference organized by Montenegro Academy of Sciences and arts. 13, pp.93–100, 2009.

• A. A. Davydov, T. S. Shutkina Optimizing a cyclic process with discount

with respect to its time average profit, RUSS MATH SURV, 2009, 64 (1), 136–138.