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

Existence and uniqueness of optimal cyclic process with Existence and uniqueness of optimal cyclic discount Tatyana process with discount Shutkina Introduction Existence theorem Tatyana Shutkina Necessary optimality condition Vladimir State University Analysis of switching shutkina@vlsu.ru function Uniqueness theorem Nonlinear control and singularities October 24-28, 2010

Contents Existence and uniqueness of optimal cyclic process with discount Tatyana Shutkina 1 Introduction Introduction 2 Existence theorem Existence theorem 3 Necessary optimality condition Necessary 4 Analysis of switching function optimality condition 5 Uniqueness theorem Analysis of switching function Uniqueness theorem

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

Arnold’s model Existence and uniqueness of optimal cyclic process with discount In a presence of a continuous profit density f on the circle an Tatyana Shutkina optimization of periodic motion could lead to the problem of the selection of cyclic process with the maximum time averaged profit: Introduction Existence � T theorem 1 f ( x ( t )) dt → max . Necessary T optimality 0 condition Analysis of V.I.Arnold shows that in a generic case the optimal strategy switching function exists, uniqueness and is rather simple. Uniqueness theorem

Problem reformulation Existence and uniqueness For an admissible motion x we introduce density of optimal cyclic ρ, ρ ( x ( t )) = 1 / ˙ x ( t ) . So almost everywhere we have dx ( t ) = ˙ x ( t ) dt process with discount or, taking into account the positiveness of admissible velocities, Tatyana dt = ρ ( x ( t )) dx ( t ) . Shutkina Thus our extremal problem could be rewritten in the form Introduction Existence 2 π 2 π theorem � � Necessary A ρ ( f ) := f ( x ) ρ ( x ) dx/ ρ ( x ) dx → max optimality condition 0 0 Analysis of switching In such a formulation we need to find a measurable density ρ on function the circle which satisfies the constraint Uniqueness theorem r 1 ≤ ρ ≤ r 2 (1)

Existence and uniqueness of optimal cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary optimality condition Analysis of switching function Uniqueness theorem

Models with discounts Analogous results were proved for the case with a positive Existence and discount σ > 0 and discount β which can be positive or negative. uniqueness of optimal cyclic 2 π 2 π process with x − σ � ρ ( z ) dz discount � � A ρ ( f ) := e f ( x ) ρ ( x ) dx/ ρ ( x ) dx → max 0 Tatyana Shutkina 0 0 Introduction or Existence theorem 2 π 2 π � � Necessary A ρ ( f ) := ( − σ ) − 1 f ( x ) d ( e − σφ ( x ) ) / d ( φ ( x )) → max . (2) optimality condition 0 0 Analysis of switching function x � where φ ( x ) = ρ ( z ) dz Uniqueness theorem 0 2 π 2 π x x − σ � ρ ( z ) dz − β � ρ ( z ) dz � � A ρ ( f ) := e f ( x ) ρ ( x ) dx/ e ρ ( x ) dx → max 0 0 0 0

Existence theorem Existence and uniqueness of optimal cyclic process with discount Tatyana Shutkina Theorem Introduction For a continuous profit density f and continuous positive Existence constraint functions r 1 , r 2 there exists a measurable density ρ max theorem which satisfies constraint r 1 ≤ ρ ≤ r 2 and provides exact upper Necessary optimality bound of values A ρ ( f ) over all measurable functions ρ complying condition with this constraint. Analysis of switching function Uniqueness theorem

Necessary optimality condition Existence and uniqueness of optimal cyclic Theorem process with discount If for a continuous profit density f and continuous positive Tatyana constraint functions r 1 , r 2 the maximum A of functional in (2) is Shutkina provided by a density ρ satisfying constraint (1) then at any point Introduction x , where ρ is derivative of its integral, the value Existence theorem 2 π y x Necessary � � − σ ρ ( z ) dz � − σ ρ ( z ) dz optimality e f ( x ) − σ e f ( y ) ρ ( y ) dy − A (3) condition 0 0 Analysis of x switching function is either non-positive or non-negative, or else zero if the value ρ ( x ) Uniqueness theorem is equal to either r 1 ( x ) or r 2 ( x ) , or else belongs to ( r 1 ( x ) , r 2 ( x )) , respectively.

Switching function Existence and Expression (3) defines a function S, S = S ( x ) . In some sense this uniqueness of optimal function plays the role of switching function. It could be rewritten cyclic process with as discount Tatyana x y Shutkina x � � − σ ρ ( z ) dz � − σ ρ ( z ) dz S ( x ) = e f ( x )+ σ e f ( y ) ρ ( y ) dy − σP − A, (4) 0 0 Introduction Existence 0 theorem Necessary y 2 π optimality − σ � ρ ( z ) dz condition � where P = e f ( y ) ρ ( y ) dy is the profit along the cycle. 0 Analysis of 0 switching Let c = − σP − A than function Uniqueness x y x theorem − σ � ρ ( z ) dz � − σ � ρ ( z ) dz S ( x ) = e f ( x ) + σ e f ( y ) ρ ( y ) dy + c 0 0 0

Existence and uniqueness of optimal cyclic process with discount Tatyana Shutkina Introduction Existence theorem Necessary optimality condition Analysis of switching function Uniqueness theorem

Uniqueness theorem Existence and uniqueness of optimal Theorem (Uniqueness theorem) cyclic process with discount For differentiable positive profit density f with a finite number of Tatyana critical points and continuous positive constraint functions r 1 , r 2 , Shutkina which coincide at isolated points only, in the presence of discount Introduction the cyclic process with maximum time averaged profit is uniquely Existence defined. theorem Necessary optimality Let m, M are the maximum and minimum values of switching condition function on zero, which means if c < m or c > M than the motion Analysis of switching in the cycle c uses only minimum r 1 or maximum r 2 density, function respectively. This values exists and for c � m (or c � M )period of Uniqueness theorem level of cycle and profit for this level of cycle are constant. It means that time average profit is constant too. The cycle for optimal average profit is cycle for some level c ∈ [ m, M ] .

The monotonicity of cycle Existence and uniqueness of optimal cyclic process with Proposition discount For continuous positive profit density f and continuous positive Tatyana Shutkina constraint functions r 1 , r 2 , which coincide at isolated points only, Introduction period of level of cycle c is continuous increase function on Existence segment [ m, M ] and has derivative theorem Necessary optimality ( r 2 ( x i ) − r 1 ( x i )) T ′ ( c ) = � condition e − σϕ ( x i ) | f ′ ( x i ) | , (5) Analysis of { x i } switching function Uniqueness without values of level, where switching function has zero on theorem critical point or the ends 0 , 2 π for zero level. x i is switching point.

The monotonicity of cycle For this formulation, it is obviously, than period is continuous. Existence and Let there be only one switching point x 1 , which is not end’s of uniqueness of optimal cycle. Switching function varies continuously with c . For cyclic process with switching functional than for small variations we doesn’t have discount other switching points. Consequently, density ρ varies on interval Tatyana Shutkina ( x 1 , x 1 + ∆ x ) Introduction △ x = −△ c −△ c Existence S ′ ( x ) + · · · = e − σϕ ( x 1 ) f ′ ( x 1 ) + . . . theorem Necessary optimality condition Analysis of switching function Uniqueness theorem

The monotonicity of cycle Existence and △ T has the same sign as △ c, because it equal uniqueness of optimal cyclic r 2 ( x 1 ) − r 1 ( x 1 ) r 1 ( x 1 ) − r 2 ( x 1 ) process with △ c + . . . or △ c + . . . , discount S ′ ( x 1 ) S ′ ( x 1 ) Tatyana Shutkina when S ′ ( x 1 ) is positive and negative, respectively. Introduction △ T = △ c r 2 ( x 1 ) − r 1 ( x 1 ) Existence e − σϕ ( x 1 ) | f ′ ( x 1 ) | + . . . theorem Necessary optimality condition Dividing last equality by △ c and taken the limit by △ c → 0 , we Analysis of get switching function dT dc ( c ) = r 2 ( x 1 ) − r 1 ( x 1 ) Uniqueness e − σϕ ( x 1 ) | f ′ ( x 1 ) | theorem This derivative is positive, if r 1 ( x 1 ) < r 2 ( x 1 ) . If there are few switching points than derivative has the same form, namely, it is just the sum of such terms.