On nonconvex optimal control problems

The paper addresses the general optimal control (OC) problem with inequality constraints and a cost functional of Bolza given by d.c. functions with respect to the state in the terminal and integrand parts of the functionals. First, we reduce the original OC problem with inequality constraints to the one without constraints with the help of the Exact Penalization Theory. Further, we show that the auxiliary (penalized) problem also possesses the state-DC-structure. Employing this property, we develop the new Global Optimality Conditions (GOCs) and discuss some its features allowing to construct the new schemes of local and global searchers. Finally we elucidate the relations of the GOCs to the classical OC theory, in particular, to the Pontryagin’s maximum principle.


Introduction
Nowadays, many optimal control (OC) problems from various application areas [1]- [5] are (implicitly or explicitly) nonconvex in the different sense. Often one can meet the nonlinear control systems that generate a huge number of local pitfalls from which it is impossible "to jump out", moreover, to reach a set of globally optimal solutions using the standard OC tools [6]- [11].
On the other hand, such objectives as a search for equilibria [17] (say, of Nash), the hierarchy structures of problem statements [4,17], finally, the inverse problem [12] from various applied fields, produce generic nonconvexities, which are difficult to overcome as to finding an approximate global solution [1]- [11], [13,14].
These situations explain the popularity of such methods, as "direct approach", B&B family [16] and "bioiniciated" family methods, which have no any mathematical foundation (in particular, the convergence results). In this paper we continue to develop the theory and methods for solving the nonconvex OC problems and, this time, with inequality constraints given by the Bolza (integro-terminal) functionals with DC-state decomposition [17]- [25].
First, we reduce the original OC Problem (P ) with constraints to Problem (P σ ) without constraints with the help of Exact Penalty Theory [27]- [32], and show that the auxiliary Problem (P σ ) possesses the DC-state property, as well [16]. The latter feature allows us to develop the socalled Global Optimality Conditions (GOCs) for the penalized Problem (P σ ) which under some assumptions become GOCs for (P ). After discussing some properties of GOCs we establish some relations with classical results, in particular, with the Pontryagin's principle [1]- [3], [6]- [13], [18,19]. largely known in the world and very powerful tool to generate numerical methods for various classes of OC problems [7]- [11].
Henceforth we will call the property of the functions f 1i (x), f i (x, u, t) to be represented as in (4)-(5), the state-DC-decomposition (representation), while the convexity properties of the functions g 1i (x), g i (x, u, t) and h 1i (x), h i (x, t) will be said the state-convexity ones [7,16,24,25].
Furthermore, taking into account, that x(·) = x(·, u), u ∈ U, is the unique solution to the system (1) corresponding to the control u(·) ∈ U, the next denotations look rather natural J i (u) := J i (x(·, u), u), i ∈ {0} ∪ I. Let all the date be smooth with respect to the state. On the other hand, thanks to the state-convexity of the functions above, in particular, the following inequalities (∇ := ∇ x ) hold true ( [7,14,15,20]) Let us now show that under the above assumptions every functional J i (u) = J i (x, u), defined in (3), can be represented in the form where G i , H i possess the state-convexity properties similar to described above. Indeed, using the notations below together with (3)-(5) we obtain the desirable state-convexity properties.
Now we are ready to study the following optimal control (OC) problem (P): Clearly, due to nonconvexity (with respect to the state) of the terminal parts f 1i and of the integrand f i (x, u, t), each functional J i (x, u), i ∈ {0} ∪ I, the feasible region of Problem (P), and Problem (P) itself, as a whole, turn out to be nonconvex. It means that Problem (P) might possess a big number of locally optimal and stationary (in the sense of PMP) processes (x * (·), u * (·)), x * (t) = x(t, u * ), t ∈ T, u * (·) ∈ U, which may be rather far from a set Sol(P) of global solutions (if one exists) even with respect to the value of the cost functional.
Hence, the existence of a threshold value σ * > 0 implies that instead of solving a sequence of unconstrained problems with σ k → ∞ [7,26] one needs to solve only a single unconstrained problem with a penalty parameter σ ≥ σ * .
In addition, it is well-known that , if a process (z(·), w(·)) is a global solution to Problem (P σ ): (z, w) ∈ Sol(P σ ), z(t) = x(t, u), t ∈ T, w ∈ U, and, besides, (z, w) is feasible in Problem (P), i.e. J i (z, w) ≤ 0, i ∈ I, then (z(·), w(·)) is a global solution to Problem (P). One has to mention that the inverse assertion, in general, does not hold.
It is worth noting that under various Constraint Qualification (CQ) conditions (e.g. MFCQ, etc.), the error bound properties, the calmness of constraint systems, etc., one can prove the existence of the Exact Penalty threshold value σ * > 0 for local and global solutions [27]- [32], [13,21]. Assume that some regularity conditions which ensure the existence of the threshold value σ * of the penalty parameter (ThrVPP) are fulfilled.
3. DC decomposition of Problem (P σ ) First, it can be readily seen that, due to the presentations (7), (7 ) and also to (9)-(11), the cost function F σ (x, u) = F σ (u) of the penalized Problem (P σ )-(10) can be represented as follows To this end, let us show now that the penalty function P nt(x, u) defined in (9) can also be represented as a DC function: P nt(x, u) = G W (x, u) − H W (x), where G W (·) and H W (·) possess the state-convexity property. Then, obviously, F σ (x, u) will be state-DC (i.e. x → F σ (x, u) is a DC function). Indeed, it follows from (9) that Now with the help of denotations one gets the following DC decomposition of the penalty function where the functions G W (·) and H W (·) obviously conserve [7,14,15] the state-convexity property in virtue of (7), (7 ) and (13). Moreover, as was claimed above, the cost function F σ (x, u), defined in (11 ), possesses the DC-state decomposition, because (see (11 ), (13), (14)) where due to (7 ) we have It is clear, thanks to (15)- (17), that the functions G σ (x, u) and H σ (x) are endowed with the state-convexity property, as well [7,13,16,18,20].
Hence, due to the state-convexity property of H σ (·), the next inequality takes place ∀u ∈ U,

Global optimality conditions (GOCs)
Let us return now to Problem (P)- (8), assuming that the feasible set is not empty, i.e.
and the optimal value V(P) of Problem (P) is finite, i.e.
Whence, with the help of the state-convexity of H σ (·) and the inequality (19), we derive the inequality (25). # Remark 1. It can be readily seen that Theorem 1 reduces the nonconvex OC Problem (P σ ) to a study of the family of the (partially) linearized problems as follows (P σ L(y)) : Φ σy (x(·, u), u) := G σ (x(·, u), u) − ∇H σ (y(·)), x(·) ↓ min u , u ∈ U, depending on the pair (y(·), β) ∈ P C(T ) × IR, fulfilling the equation (24) with the functions given by (16)- (19). It is not difficult to notice that the linearization is carried out with respect to the "united" nonconvexity of Problem (P)-(8) generated only by functionals J 0 (u), ..., J m (u) (but not by the nonlinear control system (1)-(2)), and accumulated by the function H σ (·) (see (P)- (8) and (17). Clearly, in the case when the system (1)-(2) (which is implicitly present in (26)) is state-linear (see below), then the problem (P σ L(y))- (26) becomes convex in the sense that the Pontryagin principle turns out to be the sufficient conditions for a global solution to the problem (P σ L(y))- (26). Thus, the verification of the principal inequality (25) can be reduced to the solution