QUADRATIC ORDER CONDITIONS FOR AN EXTENDED WEAK MINIMUM IN OPTIMAL CONTROL PROBLEMS WITH INTERMEDIATE AND MIXED CONSTRAINTS

. We consider a general optimal control problem with intermediate and mixed constraints. Using a natural transformation (replication of the state and control variables), this problem is reduced to a standard optimal control problem with mixed constraints, which makes it possible to obtain quadratic order conditions for an “extended” weak minimum. The conditions obtained are applied to the problem of light refraction.

The problem A contains intermediate constraints, i.e., equality and inequality constraints involving the state values not only at the endpoints of the interval [t 0 , t ν ], but also at intermediate points t 1 , t 2 , . . . , t ν−1 . Moreover, there are mixed constraints of equality type g(t, x, u) = 0. If ν = 1, i.e., there are no intermediate points, then problem A is the Lagrange problem of classical calculus of variations stated in the Pontryagin form with additional mixed equality constraints.
We assume that A1) the functions f and g are defined and continuous on an open set Q ⊂ R 1+n+r together with their partial derivatives in t, x, u up to the second order; A2) the functions ϕ i and η j are defined and twice differentiable on an open set P ⊂ R (1+ν)(1+n) ; A3) at any point (t, x, u) ∈ Q such that g(t, x, u) = 0, the matrix g u (t, x, u) has rank d.
Our aim is to obtain quadratic order conditions for a weak minimum in problem A. To this end, we reduce this problem to a standard optimal control problem without intermediate constraints.
Recall the definition of a weak minimum in problem S.
In problem A, however, the interval ∆ = [t 0 , t ν ] and all subintervals ∆ k = [t k−1 , t k ] are not fixed, so the definition of a weak minimum should be modified.
We propose first some auxiliary notions. Let f : ∆ 1 → R n be an arbitrary continuous function defined on a closed interval ∆ 1 and let be given another closed interval ∆ 2 . Extend f by a constant to the whole real line outside ∆ 1 preserving its continuity, and then take its restriction to ∆ 2 . The obtained functionf : ∆ 2 → R n will be called translation of the function f from the interval ∆ 1 to the interval ∆ 2 .
Using the notion of translation, we introduce the notion of ε− closeness of functions. Let be given a continuous function f 0 (t) defined on an interval ∆ 0 = [t 0 0 , t 0 1 ], and a number ε > 0. Definition 1.4. We will say that a measurable function f (t) defined on an interval and the translatioñ f 0 of the function f 0 to the interval ∆ satisfies the inequality |f 0 (t) − f (t)| < ε a.e. on ∆.
Define also the notion of ε− closeness of processes in problem A. For any admissible process w = (x(t), u(t), θ), t ∈ ∆ of problem A and any natural k = 1, . . . , ν, the restriction w k = (x k , u k ) of the process w on any subinterval ∆ k = [t k−1 , t k ] will be called a partial process. Definition 1.5. We will say that a process w = (x(t), u(t), θ), t ∈ ∆ is ε− close to a process w 0 = (x 0 (t), u 0 (t), θ 0 ), t ∈ ∆ 0 , where u 0 (t) is a piecewise-continuous function with possible discontinuities only at points t 0 k if all the partial processes w k are ε -close to the partial processes w 0 k , i.e. |t 0 k − t k | < ε, k = 0, . . . , ν, and the translation of any partial processw 0 k from the interval ∆ 0 k to the interval ∆ k satisfies the inequalities ũ 0 k − u k ∞ < ε, x 0 k − x k C < ε. Note that, due to the smoothness of g, definition 5 implies that, for any ε -close admissible process w = (x(t), u(t), θ), t ∈ ∆, there exists C > 0 such that all its partial processes w k satisfy the inequality | g(t,x 0 (t),ũ 0 (t)) − g(t, x(t), u(t))| < Cε for almost all t ∈ ∆. Now we can give a notion of extended weak minimum in problem A.
Thus, the base process w 0 has a piecewise-continuous control, but it is subject to comparison with all admissible ε− close processes with arbitrary measurable controls. Note that after the translation of partial processes w 0 k to ∆ k , the resulting processw 0 can fail to be admissible. The procedure of translation is used here only for estimating the closeness of the processes.
2. First and second order conditions in problem S. Recall here the known first and second order conditions for a weak minimum in problem S. Note that we consider x, u, g, ϕ, η as column vectors, while the corresponding multipliers as row vectors.
Denote the set of all such collections λ by Λ. Obviously, it is a finite-dimensional compactum, parametrized by the pair (α, β).
For any λ ∈ Λ define the Lagrange function and consider its second variation at w 0 : where all the derivatives of H(t, x, u) are taken at the point (t, x 0 (t), u 0 (t)). Define the function Ω[Λ](w) = max λ∈Λ Φ [λ](w 0 )w,w and the quadratic order These functions should be considered on the cone of critical variations Here one can consider the spaceū ∈ L r 2 [0, T ] instead of the original spaceū ∈ L r ∞ [0, T ]. This can be shown by the lemma on denseness from [6]. b) If ∃ c > 0 such that Ω[Λ](w) ≥ c γ(w) for anyw ∈ K, then the process w 0 provides a strict weak minimum in problem S.
The proof can be found in [14]. If the endpoint inequalities are absent, and Λ consists of a single collection λ, this theorem is a well known fact of classical calculus of variations, see e.g. [3]. Note that in part b) one can actually assert more [14]: ∃ c > 0 and ε > 0 such that, for any pair (x, u) satisfying the inequalities

First order conditions in problem A.
To obtain optimality conditions in problem A, we use a quite natural trick, in fact, a change of variables, which was used earlier by a number of authors, both for obtaining optimality conditions (probably, the first work was [5], see also [16,4,2,7,8]) and for constructing numerical algorithms (see e.g. [12,17]). Recently, it was used for obtaining quadratic order conditions for an extended weak minimum in problem A without mixed constraints [11].
be an admissible process in problem A. Following [7,8], introduce a new time τ ∈ [0, 1] and define the functions The function ρ k plays the role of time t on the interval ∆ k = [t k−1 , t k ], while z k is the length of ∆ k .
Obviously, z k (τ ) and ρ k (τ ) satisfy the equations Define the functions which satisfy the relations: where, for the sake of brevity, we set Over the set of all processesw = (z(τ ), ρ(τ ), y(τ ), v(τ )) satisfying the constraints (3)- (7), let us minimize the cost functional The obtained optimal control problem will be called problemÃ. Here the state variables are z k , ρ k and y k , while the controls are v k , k = 1, . . . , ν; the time The open set P consists of all vectorsp, for which the "truncated" vectorp ∈ P.
Let us find the correspondence between the admissible processes in problems A andÃ.
Using the relations one can as well define the inverse mapping G = F −1 , which also preserves the value of the cost. The constructed mappings F and G possess the following properties.
is a piecewisecontinuous function with possible discontinuities at points t 0 k , provides an extended weak minimum in problem A, then the processw 0 = F (w 0 ) provides a weak minimum in problemÃ, and vice versa, if a processw 0 = (z 0 (τ ), ρ 0 (τ ), y 0 (τ ), v 0 (τ )) with continuous controls v 0 k provides a weak minimum in problemÃ, then the process w 0 = G(w 0 ) provides an extended weak minimum in problem A.
The proof is similar to the proof of Theorem 2 in [11] related to problem A without mixed constraints.
Thus, the study of the extended weak minimality of a process w 0 in problem A is reduced to the study of the weak minimality of the processw 0 = F (w 0 ) in problemÃ.
; e) transversality at the endpoints:

ANDREI V. DMITRUK AND ALEXANDER M. KAGANOVICH
f ) jump conditions for ψ x and ψ t at the intermediate points: We claim that actually one can write This can be obtained by introducing more intermediate points t j , j = 1, . . . , N, including the "old" t k , so that the new intervals ∆ j = [t j−1 , t j ] are arbitrarily small. Then the integral relations ∆ j H(t, x 0 (t), u 0 (t)) dt = 0 for all ∆ j readily yield (20). Since the control u 0 (t) is continuous on every "old" interval ∆ k , the notion of extended weak minimum would not change. The jump conditions at all additional t j would only say that ψ x and ψ t are continuous at these points, i.e. no new conditions to e) and f) would appear.
4. Quadratic order conditions in problemÃ . Denote by π = (z, ρ, y, v) the quadruple of variables in problemÃ, and byπ = (z,ρ,ȳ,v) its variation. Letq be a variation ofp. The second variation of the Lagrange function in problemÃ at the pointw 0 is Consider the functionalΩ and define the quadratic order These functions should be considered on the cone of critical variations ϕ i (p 0 )q ≤ 0, i ∈ I, η j (p 0 )q = 0, j = 1, . . . , q, where all the derivatives of f and g are taken at (ρ 0 k (τ ), y 0 k (τ ), v 0 k (τ )), and I is the set of active indices.
In this notation, the second variation takes the form (here we took into account that H zz = 0). Using formula (9), rewrite the last expression as where Π k (µ k ) = Π k (µ k ) −m k g(µ k ), and its derivatives are taken at the point µ 0 k (τ ). Now, consider the process w 0 = (x 0 , u 0 ) = G(w 0 ) of problem A, corresponding to the processw 0 , that is defined by formulas (16), and the Lagrange multipliers λ = (α, β, ψ x , ψ t , m), defined by formulas (17). Define the variations and rewrite the coneK, the functionalΩ, and the orderγ in these new variables.

ANDREI V. DMITRUK AND ALEXANDER M. KAGANOVICH
Thus, the cone K in the new variables has the form: x(t) andt(t) are continuous on the whole ∆ 0 , Then, the variation of process w 0 can be represented asw(t) =μ(σ 0 k (t)), k = 1, . . . ν, and so, the k -th term in the integral part of (21) can be written as where all the derivatives of H are taken at the point (t, x 0 (t), u 0 (t)). The stationarity condition yields H u [λ](t) = H v k [λ](τ (t)) = 0 for all t ∈ [t 0 0 , t 0 ν ]. Thus, the quadratic form in the new variables is Note that variationsz k come in the cone and the quadratic form with the multiplier 1/z 0 k . Making the changez k =z k /z 0 k , we obtain an equivalent quadratic order γ , so the fractionz k /z 0 k in the cone K and the quadratic form Ω can be harmlessly changed by the termz k , which we will still denote byz k . d) Order γ. In the new variables, the quadratic orderγ on the cone K takes the form: Sincex satisfies on ∆ 0 k an ODE that is linear inz k ,t,x,ū and the functiont(t) is linear in t, we get the estimate hence, the terms |x(t 0 k )| 2 , k = 1, . . . , ν − 1, can be excluded from γ, remaining only |x(t 0 0 )| 2 (or any one |x(t 0 k0 )| 2 , excluding all other terms with k = k 0 ).
For any k = 1, . . . , ν the value oft(t) at t 0 k can be represented as Therefore, |t(t k )| ≤ const (|t(t 0 )| + k r=1 |z r |), and so, the termst 2 (t 0 k ), k = 1, . . . , ν − 1 can also be excluded from γ, remaining onlyt 2 (t 0 0 ) (or any one |t(t 0 k0 )| 2 , excluding all other terms with k = k 0 ). Thus, the order γ on K can be taken in the form 5. Quadratic conditions for problem A. The above analysis of conditions for problemÃ gives the following result for problem A. Let a process w 0 = (x 0 (t), u 0 (t), θ 0 ), t ∈ [t 0 0 , t 0 ν ], where u 0 (·) is a piecewise continuous function with possible discontinuities at points t 0 k , satisfy the EL equation for problem A (see theorem 3.3). Let Λ be the corresponding set of collections of normalized Lagrange multipliers λ = (α, β, ψ x , ψ t , m), where α = (α 0 , . . . , α h ), . Define the cone of critical variations . . , ν, t(t) andx(t) are continuous on the whole ∆ 0 , g 0 tt (t) + g 0 xx (t) + g 0 uū (t) = 0 a.e. on ∆ 0 , ϕ i (p 0 )p ≤ 0, i ∈ I, η j (p 0 )p = 0, j = 1, . . . , q , where I is the set of active indices, and the derivatives of f and g are taken at the point (t, x 0 (t), u 0 (t)). For any λ ∈ Λ andw = (t,x,ū) define the quadratic form  b) If ∃ c > 0 such that Ω[Λ](w) ≥ c γ(w) for anyw ∈ K, then the process w 0 provides a strict extended weak minimum in problem A. (Moreover, the corresponding violation function satisfies the below estimate. We do not formulate it in detail.) Remark 1. Problem A is a generalization of the canonical problem S. Let us show that, for a piecewise-continuous control u 0 (t), the quadratic order conditions for an extended weak minimum in problem A (theorem 5.1) generalize the quadratic order conditions in the canonical problem S (see e.g. [14]).
Since the interval [t 0 , t 1 ] in problem S is fixed, we havet(t 0 ) = 0 andt(t 1 ) = 0. In view of linearity of the functiont(t) we then obtaint(t) ≡ 0, hence alsoz = 0. One can easily check that theorem 5.1 fort =z = 0 transforms into the quadratic order conditions in problem S.
Thus, for a piecewise-continuous control u 0 (t), theorem 5.1 can be considered as a generalization of the quadratic order conditions for a weak minimum in problems with mixed constraints to problems with mixed and intermediate constraints.

Remark 2.
In [1] the authors consider a problem of type A with mixed constraints but nonvariable intermediate times t k . The proposed quadratic necessary conditions are: Ω[Λ h+q ](w) ≥ 0 on K, where Λ h+q is the set of λ ∈ Λ such that the index of quadratic form Ω[λ](w) (the dimension of a subspace on which it is negative) is not greater than h + q.
The paper [11] deals with a problem of type A without mixed constraints. For this class of problems, the obtained results are equivalent to theorem 5.1 (since, in the absence of mixed constraints, we have H u = 0 ). Because of this, theorem 5.1 can be considered as a generalization of the quadratic order conditions for an extended weak minimum in problems with intermediate constraints to problems with intermediate and mixed equality type constraints.
In [15] and [14,Ch.2], Osmolovskii considered the so-called broken extremals in problems of calculus of variations. It is a particular case of problem A, where the control system isẋ = u, the mixed constraints are absent, and the functions ϕ, η depend only on the endpoints and do not depend on the intermediate points (so, the last ones only mark the intervals of continuity of u 0 (t) ). The author obtains quadratic conditions of a θ− minimum which is just a bit stronger than our extended weak minimum. Though the results have rather different form than theorem 5.1, we believe that on the class of mutual applicability they are equivalent. A more detailed comparison will be given in a later paper.
Remark 3. The above transformation (replication of state and control variables) can be also used for obtaining quadratic order conditions for an extended weak minimum in the following generalizations of problem A. a) Consider a problem of type A in which, on every subinterval ∆ k , the process should satisfy its own mixed constraints: g k s (t, x, u) = 0, s = 1, . . . , d k (problem A ). Obviously, this statement does not satisfy Assumption A3 on the whole interval ∆.
Note that, in derivation of Euler-Lagrange equation, we needed the fulfilment of Assumption A3 only on every subinterval ∆ k , because each mixed constraint g s = 0 in problemÃ decomposes into ν mixed constraints in accordance with the The other conditions of theorem 5.1 come to theorem 5.2 without changes. c) As a further generalization of problem A consider a problem coinciding in its form with problem B, but having on each ∆ k its own control systemẋ k = f k (t, x k , u k ), where x k ∈ R n k , u k ∈ R r k of its own dimensions (problem C ). In the case of coinciding dimensions of x on neighboring subintervals ∆ k this statement allows for discontinuities of the trajectory at the points t k .
Applying the procedure of obtaining quadratic order conditions, similar to that used for problem B (here we do not need to replicate the variables x, u, because from the outset, they are different for each ∆ k and it only remains to redefine them on a common time interval), one can obtain conditions for an extended weak minimum also in problem C (theorem 5.3).
The difference of theorem 5.3 from theorem 5.2 is that now the cone K includes, instead of one functionx(t), a tuple of functionsx k (t) ∈ R n k defined on their respective intervals ∆ 0 k , k = 1, . . . , ν, each satisfying its own differential equation. Moreover, since the original trajectory admits discontinuities, no junction conditions at the points t k forx k will appear on the cone K. In the order γ, the term |x(t 0 0 )| 2 should be replaced by The other conditions of theorem 5.2 come to theorem 5.3 without changes.

Example.
To demonstrate the application of the obtained quadratic order conditions, consider the classical problem on the refraction of the light ray.
Let in the space R n there be given two isotropic optical media separated by a smooth surface S = {x ∈ R n : g(x) = 0} without singular points. The speed of light in each medium is constant. A light ray emanates from a point x 0 in the first medium and comes to a point x 2 in the second medium. Since the media are different, the light ray breaks at a point x 1 where it intersects the surface S. According to the Fermat's principle, the trajectory from x 0 to x 2 corresponds to a minimal time. It is required to find the trajectory.
This problem can be stated as the following time-optimal control problem with an intermediate and a mixed equality type constraints (like e.g. in [7]): where g(a) < 0, g(b) > 0, the moments 0 < t 1 < t 2 are not fixed, and c k > 0, k = 1, 2 are given speeds of light in the both media. (Since we use the scalar product, we identify the column and row vectors.) Here the Pontryagin function is the extended Pontryagin function is and the terminal Lagrange function is where the collection (α 0 , β t0 , β x0 , β x2 , δ) is nontrivial. Let us write out the Euler-Lagrange equation.
So, we can set α 0 = 1 and determine uniquely all the multipliers. Indeed, ψ t ≡ −1, m(t) ≡ 1, and (28) yields This implies that u is a piecewise-constant function with a possible discontinuity at the point t 1 , and since |u| = 1, it can be represented in the form: Calculating the duration of motion on each interval ∆ k as the ratio between the distance and the speed, we obtain The unknown vector x(t 1 ) can be found from the system of equations g(x(t 1 )) = 0, where ψ x (t 1 ± 0) are expressed through x(t 1 ) by the formulas (29) and (30).
In the generic case this system has a solution which determines the whole extremal. Thus, the obtained trajectory of the light is a concatenation of two straight line segments with a common point at the boundary of the given two media.
(This fact can be as well obtained without an advanced theory, but the condition (u 2 /c 2 − u 1 /c 1 ) || g (x(t 1 ) require some optimization arguments.) Now, let us write out the quadratic order conditions of an extended weak minimum for the found extremal.
On this subspace, the quadratic form is and the quadratic order is γ(w) =z 2 1 +z 2 2 + |x(t 1 )| 2 + t2 0 |ū(t)| 2 dt (here in γ we take for convenience |x(t 1 )| 2 ) instead of |x(t 0 )| 2 ). Let us analyze the quadratic order conditions. The order γ containsz 2 1 andz 2 2 , but Ω does not depend explicitly onz 1 andz 2 . We claim that bothz 2 k ≤ const |x(t 1 )| 2 . Indeed, among the relations on K we have the differential equatioṅ Multiply scalarly both sides of this equation by u and then integrate over the interval ∆ k . Since the function u(t) = u k is constant on each ∆ k , moreover, |u| = 1 and (u,ū) = 0 on K, we get In view of the endpoint relationsx(0) =x(t 2 ) = 0, these equalities give which implies that bothz 2 k ≤ const |x(t 1 )| 2 , so the claim is proved. The obtained estimates allow us to exclude the termsz 2 1 andz 2 2 from γ. Then, the quadratic order can be taken in a reduced form: Thus, to verify the extended weak minimality of the given process, one has to check the sign definiteness of quadratic form (34) with respect to γ on the subspace K defined in (33). Looking at (34), one can notice that the first part of Ω makes this task not obvious, in general. This depends on the specificity of g(x). Let us consider the following