Handbook of mathematics for engineers and scienteists part 147 pot

990 SPECIAL FUNCTIONS AND THEIR PROPERTIES 18.18.2-2. Generating function. Fourier series expansions. Integrals. The generating function is expressed as 2e xt e t + 1 ≡ ∞  n=0 E n (x) t n n! (|t| < π). This relation may be used as a definition of the Euler polynomials. Fourier series expansions: E n (x)=4 n! π n+1 ∞  k=0 sin  (2k + 1)πx – 1 2 πn  (2k + 1) n+1 (n = 0, 0 < x < 1; n > 0, 0 ≤ x ≤ 1); E 2n (x)=4(–1) n (2n)! π 2n+1 ∞  k=0 sin  (2k + 1)πx  (2k + 1) 2n+1 (n = 0, 0 < x < 1; n > 0, 0 ≤ x ≤ 1); E 2n–1 (x)=4(–1) n (2n – 1)! π 2n ∞  k=0 cos  (2k + 1)πx  (2k + 1) 2n (n = 1, 2, , 0 ≤ x ≤ 1). Integrals:  x a E n (t) dt = E n+1 (x)–E n+1 (a) n + 1 ,  1 0 E m (t)E n (t) dt = 4(–1) n (2 m+n+2 – 1) m! n! (m + n + 2)! B m+n+2 , where m, n = 0, 1, and B n are Bernoulli numbers. The Euler polynomials are orthog- onal for even n + m. Connection with the Bernoulli polynomials: E n–1 (x)= 2 n n  B n  x + 1 2  – B n  x 2   = 2 n  B n (x)–2 n B n  x 2   , where n = 1, 2, References for Chapter 18 Abramowitz,M.andStegun,I.A.(Editors),Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics, Washington, D.C., 1964. Bateman, H. and Erd ´ elyi, A., Higher Transcendental Functions, Vol. 1 and Vol. 2, McGraw-Hill, New York, 1953. Bateman, H. and Erd ´ elyi, A., Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York, 1955. Gradshteyn,I.S.andRyzhik,I.M.,Tables of Integrals, Series, and Products, Academic Press, New York, 1980. Magnus, W., Oberhettinger, F., and Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd Edition, Springer-Verlag, Berlin, 1966. McLachlan, N. W., Bessel Functions for Engineers, Clarendon Press, Oxford, 1955. Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2003. Slavyanov, S. Yu. and Lay, W., Special Functions: A Unified Theory Based on Singularities, Oxford University Press, Oxford, 2000. Weisstein, E. W., CRC Concise Encyclopedia of Mathematics, 2nd Edition, CRC Press, Boca Raton, 2003. Zwillinger, D., CRC Standard Mathematical Tables and Formulae, 31st Edition, CRC Press, Boca Raton, 2002. Chapter 19 Calculus of Variations and Optimization 19.1. Calculus of Variations and Optimal Control 19.1.1. Some Definitions and Formulas 19.1.1-1. Notion of functional. Let a class M of functions x(t) be given. If for each function x(t) M there is a certain number J assigned to x(t) according to some law, then one says that a functional J = J[x] is defined on M. Example 1. LetM =C 1 [t 0 , t 1 ] betheclassof functions x(t)defined on the interval[t 0 , t 1 ] and continuously differentiable on this interval. Then J[x]=  t 1 t 0  1 +[x  t (t)] 2 dt is a functional defined on this class of functions. Geometrically, this functional expresses the length of the curve x = x(t) with endpoints A(t 0 , x(t 0 )) and B(t 1 , x(t 1 )). Calculus of variations established conditions under which functionals attain their ex- trema. Suppose that a functional J = J[x] attains its minimum or maximum at a function ˆx. A strong (zero-order) neighborhood of ˆx is the set of continuous comparison functions (or trial functions) x such that |x(t)–ˆx(t)| < ε (t 1 ≤ t ≤ t 2 ) for a given ε > 0.Aweak (first-order) neighborhood of ˆx is the set of piecewise continuous comparison functions x such that |x(t)–ˆx(t)| + |x  t (t)–ˆx  t (t)| < ε (t 1 ≤ t ≤ t 2 ) for a given ε > 0.Ifˆx(t) minimizes the functional J[x] in a strong (resp., weak) neigh- borhood of itself, then it is called a point of strong (resp.,weak) minimum of the functional J[x]. If ˆx maximizes the functional J[x] in a strong (resp., weak) neighborhood of itself, then it is called a point of strong (resp., weak) maximum of the functional J[x]. Any strong extremum is also a weak extremum. Strong and weak extrema are relative extrema.The extremum of the functional J[x] over the entire domain where it is definediscalledan absolute extremum. An absolute extremum is also a relative extremum. The function classes conventionally used in calculus of variations: 1. The class C[t 0 , t 1 ] of continuous functions on the interval [t 0 , t 1 ] with the norm x(t) 0 =max t 1 ≤t≤t 2 |x(t)|. 2. The class C 1 [t 0 , t 1 ] of functions with continuous first derivative on the interval [t 0 , t 1 ] with the norm x(t) 1 =max t 1 ≤t≤t 2 |x(t)| +max t 1 ≤t≤t 2 |x  t (t)|. 991 992 CALCULUS OF VARIATIONS AND OPTIMIZATION 3. The class C n [t 0 , t 1 ] of functions with continuous nth derivative on the interval [t 0 , t 1 ] with the norm x(t) n = n  k=1 max t 1 ≤t≤t 2 |x (k) t (t)|. When stating variational problems, one should indicate which kind of extremum is to be found and specify the function class in which it is sought. Remark 1. The class PC 1 [t 0 , t 1 ] of continuous functions with piecewise continuous derivative on the interval [t 0 , t 1 ] is equipped with the norm indicated in item 1. The class PC n [t 0 , t 1 ] of continuous functions with piecewise continuous nth derivative on the interval [t 0 , t 1 ] is equipped with the norm indicated in item 2. Remark 2. We do not specify function classes whenever the results are valid for an arbitrary normed space. 19.1.1-2. First variation of functional. The difference δx = x – x 0 (19.1.1.1) of two functions x(t)andx 0 (t) in a given function class M is called the variation (or increment) of the argument x(t) of the functional J[x]. The difference ΔJ ≡ ΔJ[x]=J[x + δx]–J[x](19.1.1.2) is called the increment of the functional J[x] corresponding to the increment δx of the argument. First definition of the variation of a functional. If the increment (19.1.1.2) can be represented as ΔJ = L[x, δx]+β[x, δx]δx,(19.1.1.3) where L[x, δx] is a functional linear in δx and β[x, δx] → 0 as δx →0, then the linear part L[x, δx] of the increment of the functional is called the variation of the functional and is denoted by δJ. In this case, the functional ΔJ[x]issaidtobedifferentiable at the point x(t). Second definition of the variation of a functional. Consider the functional F(α)= J[x + αδx]. The value δJ = F  α (0)(19.1.1.4) of its derivative with respect to the parameter α at α = 0 is called the variation of the functional J[x] at the point x(t). If the variation of a functional exists as the principal linear part of its increment, i.e., in the sense of the first definition, then the variation understood as the value of the derivative with respect to the parameter α at α = 0 also exists and these definitions coincide. Remark. The second definition of the variation of a functional is somewhat wider than the first: there exist functionals whose increments do not have principal linear parts, but their variations in the sense of the second definition still exist. 19.1.1-3. Second variation of functional. A functional J[x, y] depending on two arguments (that belong to some linear space) is said to be bilinear if it is a linear functional of each of the arguments. If we set y ≡ x in a bilinear functional, then the resulting functional J[x, x]issaidtobequadratic. A bilinear functional in a finite-dimensional space is called a bilinear form. 19.1. CALCULUS OF VARIATIONS AND OPTIMAL CONTROL 993 A quadratic functional J[x, x]issaidtobepositive definite if J[x, x]>0 for all x ≠ 0. A quadratic functional J[x, x]issaidtobestrongly positive if there is a constant C > 0 such that J[x, x] ≥ Cx 2 for all x. Example 2. The integral  t 1 t 0  t 1 t 0 K(s, t)x(s)y(t) ds dt, where K(s, t)isafixed function of two variables, is a bilinear functional on C[t 0 , t 1 ]. First definition of the second variation of a functional. If the increment (19.1.1.2) of a functional can be represented as ΔJ = L 1 [δx]+ 1 2 L 2 [δx]+βδx 2 ,(19.1.1.5) where L 1 [δx] is a linear functional, L 2 [δx] is a quadratic functional, and β →0 as δx→0, then the quadratic functional L 2 [δx] is called the second variation (second differential) of the functional J[x] and is denoted by δ 2 J. Second definition of the second variation of a functional. Consider the functional F(α)=J[x + αδx]. The value δ 2 J = F  αα (0)(19.1.1.6) of its second derivative with respect to the parameter α at α = 0 is called the second-order variation of the functional J[x] at the point x(t). The first and second variations of a functional permit stating necessary conditions for the minimum or maximum of a functional. Necessary conditions for the minimum or maximum of a functional: 1. The first variation must be zero, δJ = 0 . 2. The second-order variation must be nonnegative, δ 2 J ≥ 0 , in the case of minimum; and nonpositive, δ 2 J ≤ 0 , in the case of maximum. 19.1.2. Simplest Problem of Calculus of Variations 19.1.2-1. Statement of problem. The extremal problem J[x] ≡  t 1 t 0 L(t, x, x  t ) dt →extremum, x = x(t), x(t 0 )=x 0 , x(t 1 )=x 1 (19.1.2.1) is called the simplest problem of calculus of variations. The function L(t, x, x  t ) is called the Lagrangian, and the functional J is referred to as a classical integral functional. One usually assumes that the Lagrangian is jointly continuous in the arguments and has continuous partial derivatives of order ≤ 3. The extremum in the problem is sought in the set of continuously differentiable functions x(t) C 1 [t 0 , t 1 ] on the interval [t 0 , t 1 ] satisfying the boundary (or endpoint) conditions x(t 0 )=x 0 and x(t 1 )=x 1 . Such functions are said to be admissible. An admissible function ˆx(t) provides a weak local minimum (resp., maximum) in prob- lem (19.1.2.1) if it provides a local minimum (resp., maximum) in the space C 1 [t 0 , t 1 ], i.e., if there exists a δ > 0 such that the inequality J[x] ≥ J[ˆx] (resp., J[x] ≤ J[ˆx]) (19.1.2.2) holds for any admissible function x(t) C 1 [t 0 , t 1 ] such that x(t)–ˆx(t) 1 < δ. 994 CALCULUS OF VARIATIONS AND OPTIMIZATION The classical calculus of variations deals not only with weak extrema, but also with strong extrema. In the latter case, one considers functions x(t) in the class PC 1 [t 0 , t 1 ]; i.e., the extremum is sought in the class of piecewise continuously differentiable functions satisfying the endpoint conditions. An admissible function ˆx(t) PC 1 [t 0 , t 1 ] provides a strong local minimum (resp., maximum) in problem (19.1.2.1) if there exists a δ > 0 such that the inequality J[x] ≥ J[ˆx] (resp., J[x] ≤ J[ˆx]) (19.1.2.3) holds for any admissible function x(t) PC 1 [t 0 , t 1 ] such that x –ˆx 0 < δ. Since the set of admissible functions is wider for a strong extremum than for a weak extremum, the following assertion is true: if ˆx(t) C 1 [t 0 , t 1 ] provides a strong extremum, then it also provides a weak extremum. It follows that, for such functions, a necessary condition for weak extremum is a necessary condition for strong extremum, and a sufficient condition for strong extremum is a sufficient condition for weak extremum. 19.1.2-2. First-order necessary condition for extremum. Euler equation. For the simplest problem of calculus of variations, the variation δJ of the classical func- tional J[x]hastheform δJ[x, h]=  t 1 t 0  ∂L ∂x – d dt ∂L ∂x  t  hdt,(19.1.2.4) where h(t) is an arbitrary smooth function satisfying the conditions h(t 0 )=h(t 1 )=0 and d dt = ∂ ∂t + x  t ∂ ∂x + x  tt ∂ ∂x  t . A necessary condition for admissible function x(t) to provide a weak extremum in problem (19.1.2.1) is that δJ = 0, i.e., that the function x(t) satisfies the Euler equation ∂L ∂x – d dt ∂L ∂x  t = 0.(19.1.2.5) Here we assume that the functions L, ∂L/∂x, ∂L/∂x  t are continuous as functions of three variables (t, x,andx  t ), and ∂L/∂x  t C 1 [t 0 , t 1 ]. The solutions of the Euler equa- tion (19.1.2.5) are called extremals. Admissible functions satisfying the Euler equation are called admissible extremals. The Euler equation (19.1.2.5) in expanded form reads L x – L tx  t – x  t L xx  t – x  tt L x  t x  t = 0.(19.1.2.6) From now on, the subscripts t, x,andx  t indicate the corresponding partial derivatives. If L x  t x  t ≠ 0, then equation (19.1.2.6) is a second-order differential equation, so that its general solution depends on two arbitrary constants. The values of these constants are in general determined by the boundary conditions x(t 0 )=x 0 and x(t 1 )=x 1 . The boundary value problem L x – d dt L x  t = 0, x = x(t), x(t 0 )=x 0 , x(t 1 )=x 1 does not always have a solution; if a solution exists, it may be nonunique. 19.1. CALCULUS OF VARIATIONS AND OPTIMAL CONTROL 995 The Euler equation (19.1.2.5) for a classical functional is a second-order differential equation, and so every solution x(t) of this equation must have the second derivative x  tt (t). But sometimes a function on which a classical functional J[x] attains an extremum is not twice differentiable. An extremal also satisfies the equation d dt  L – x  t L x  t  – L x = 0.(19.1.2.7) The points of an extremal x(t)atwhichL x  t x  t ≠ 0 are said to be regular. If all points of an extremal are regular, then the extremal itself is said to be regular (or nonsingular).For regular extremals, the Euler equation can be reduced to the form x  tt = f (t, x, x  t ). Remark 1. An extremal x(t) can have a break point only if L x  t x  t = 0. Remark 2. Extremals, i.e., functions “suspected for extremum,” should not be confused with functions that actually provide an extremum. 19.1.2-3. Integration of Euler equation. 1 ◦ . The Lagrangian L is independent of x  t ; i.e., L(t, x, x  t ) ≡ L(t, x). The Euler equation (19.1.2.5) becomes L x = 0.(19.1.2.8) In general, the solution of this finite equation does not satisfy the boundary conditions x(t 0 )=x 0 and x(t 1 )=x 1 . There exists an extremal only in the exceptional cases in which the curve (19.1.2.8) passes through the boundary points (t 0 , x 0 )and(t 1 , x 1 ). 2 ◦ . The Lagrangian L depends on x  t linearly; i.e., L(t, x, x  t ) ≡ M(t, x)+N(t, x)x  t . The Euler equation (19.1.2.5) becomes M x – N t = 0,(19.1.2.9) where derivative M x and function N are evaluated at x = x(t). In general, the curve determined by equation (19.1.2.9) does not satisfy the boundary conditions, and hence, as a rule, the variational problem does not have a solution in the class of continuous functions. If equation (19.1.2.9) is satisfied identically in a domain D on the plane OXY ,then L(t, x, x  t ) dt = M(t, x) dt + N(t, x) dx is an exact differential; consequently, J[x] is inde- pendent of the integration path and has a constant value for all x. In this case, the variational problem becomes meaningless. 3 ◦ . The Lagrangian L is independent of x; i.e., L(t, x, x  t ) ≡ L(t, x  t ). The Euler equation (19.1.2.5) becomes d dt L x  t = 0,(19.1.2.10) whence it follows that L x  t = const. (19.1.2.11) Equation (19.1.2.11) is a first-order differential equation. By integrating this equation, we obtain the extremals of the problem. 996 CALCULUS OF VARIATIONS AND OPTIMIZATION Remark. Relation (19.1.2.11) is called the momentum conservation law. 4 ◦ . The Lagrangian L is independent of t; i.e., L(t, x, x  t ) ≡ L(x, x  t ). The Euler equation (19.1.2.6) becomes L x – x  t L xx  t – x  tt L x  t x  t = 0,(19.1.2.12) whence we readily obtain a first integral of the Euler equation: L – x  t L x  t = const. (19.1.2.13) Remark. Relation (19.1.2.13) is called the energy conservation law. 5 ◦ . The Lagrangian L depends only on x  t ; i.e., L(t, x, x  t ) ≡ L(x  t ). The Euler equation (19.1.2.6) becomes x  tt L x  t x  t = 0.(19.1.2.14) In this case, the extremals are given by the equations x(t)=C 1 t + C 2 ,(19.1.2.15) where C 1 and C 2 are arbitrary constants. Example 1. Let us give an example in which there exists a unique admissible extremal that provides a global extremum. Let J[x]=  1 0 (x  t ) 2 dt → min, x = x(t), x(0)=0, x(1)=1. The Euler equation becomes 2x  tt = 0. The extremals are given by the equation x(t)=C 1 t + C 2 . The unique admissible extremal is the function ˆx(t)=t, which provides the global minimum. Suppose that x  t (t) C 1 [0, 1], x(0)=0, x(1)=1.Then J[x]=J[ˆx + h]=  1 0 (1 + h  t ) 2 dt = J[ˆx]+  1 0 (h  t ) 2 dt ≥ J[ˆx] for an arbitrary function h(t) such that h(0)=0 and h(1)=0. Example 2. Let us give an example in which there exists a unique admissible extremal that provides a weak extremum but does not provide a strong extremum. Let J[x]=  1 0 (x  t ) 3 dt → min; x = x(t), x(0)=0, x(1)=1. The Euler equation becomes 6x  t x  tt = 0. The extremals are given by the equation x(t)=C 1 t + C 2 . The unique admissible extremal is the function ˆx(t)=t, which provides a weak local minimum. Indeed, J[x]=J[ˆx + h]=  1 0 (1 + h  t ) 3 dt = J[ˆx]+  1 0 (h  t ) 2 (3 + h  t ) dt for an arbitrary function h(t) such that h(0)=h(1)=0. Thus, if h 1 ≤ 3,then3 +h  t > 0 and hence J[x] ≥ J[ˆx]. Consider the sequence of functions g n (t)=  – √ n, t [0, 1/n), 0, t [1/n, 1/2], 2/ √ n, t (1/2, 1]. h n (t)=  t 0 g n (τ )dτ (n = 2, 3, ). Obviously, h n (0)=h n (1)=0 and h(t) 0 → 0 as n →∞.Wesetx n (t)=h  t (t)+h n (t) and obtain a sequence of functions x n (t) such that x n (0)=0, x n (1)=1, x n (t) → ˆx(t)inC[0, 1], and J[x n ]=J[ˆx + h n ]=  1 0 (1 + h  t ) 3 dt = 1 + 3  1 0 g 2 n dt +  1 0 g 3 n dt = 1 +  1/n 0  3n – n 3/2  dt +  1 1/2  12 n – 8 n √ n  dt =– √ n + O(1). In this case J[x n ] → –∞ as n →∞. . References for Chapter 18 Abramowitz,M.andStegun,I.A.(Editors) ,Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics, . 1955. Gradshteyn,I.S.andRyzhik,I.M.,Tables of Integrals, Series, and Products, Academic Press, New York, 1980. Magnus, W., Oberhettinger, F., and Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical. 1966. McLachlan, N. W., Bessel Functions for Engineers, Clarendon Press, Oxford, 1955. Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition,