On the Monge problem and multidimensional optimal control

Using new results on the general Monge parametrization (see [25] and the references therein) recently obtained in [21], i.e., on the possibility to extend the concept of image representation to non-controllable multidimensional linear systems, we show that we can transform some quadratic variational problems (e.g., optimal control problems) with differential constraints into free variational ones directly solvable by means of the standard Euler-Lagrange equations. This result generalizes for non-controllable multidimensional linear systems the results obtained in [11], [19] for controllable ones. In particular, in the 1-D case, this result allows us to avoid the controllability condition commonly used in the behavioural approach literature for the study of optimal control problems with a finite horizon and replace it by the stabilizability condition for the ones with an infinite horizon. Keywords— Monge problem, parametrizability, multidimensional optimal control, variational problems, controllability, autonomous elements, stabilizability. I. I NTRODUCTION Let D = A[∂1, . , ∂n] be a ring of differential operators with coefficients in the differential ringA (e.g., A = R, R[x1, . . . , xn], R(x1, . . . , xn)), ∂i = ∂/∂xi. Moreover let R ∈ Dq×p be a matrix of differential operators and F a left D-module (e.g.,C∞(Rn)), namely, it satisfies: ∀ P1, P2 ∈ D, ∀ y1, y2 ∈ F : P1 y1 + P2 y2 ∈ F . A linear system of partial differential equations (PDEs) is then defined by: kerF (R.) , {η ∈ F | R η = 0}. kerF (R.) is calledbehaviourin the behavioural approach to multidimensional linear systems ([11], [13], [24]). The classical Monge problem questions the existence of a matrix of differential operatorsQ ∈ Dp×m which satisfies: kerF (R.) = imF (Q.) , QF. See [25] for more historical details. If such a matrix Q exists, then we say that Q is a parametrization of the systemkerF (R.). In the behavioural approach to multidimensional linear systems, we say that the behaviour kerF (R.) admits an image representation ([11], [13], [18], [24]). We refer to [3], [22] for an introduction to the Monge problem and new results. It A. Quadrat is with INRIA Sophia Antipolis, CAFE project, 200 4 Route des Lucioles BP 93, 06902 Sophia Antipolis Cedex, Fran ce, Alban.Quadrat@sophia.inria. fr. D. Robertz is with Lehrstuhl B f̈ur Mathematik, RWTH Aachen, Templergraben 64, 52056 Aachen, Germany, daniel@momo.math.rwth-aachen.de. was shown by Pillai and Shankar that the existence of a parametrization of a multidimensional linear system defined by PDEs with constant coefficients is equivalent to the C∞-controllability of the system in terms of the possibility to patch two solutions ([11]). This last result extends forn-D linear systems with constant coefficients a result of J. C. Willems obtained for 1-D linear systems ([13]). See also [14], [16], [17]. Multidimensional optimal control theory has recently been developed in [11], [14], [19], [20]. Let us recall one of the main results. We refer to [11], [14], [19], [20] for more mathematical information. Theorem 1 ([11], [14], [19], [20]): Let R ∈ Dq×p, F a left D-module andkerF (R.) = {η ∈ Fp | R η = 0} a linear system of PDEs of order r. Let us suppose that we have the following parametrization η = Qξ, ∀ ξ ∈ F, of the systemkerF (R.), i.e., kerF (R.) = QFm, where Q ∈ Dp×m. Let us consider the problem of extremizing the quadratic cost I = ∫ 1 2 η r Lηr dx, where: ηr = (∂ α η = ∂1 1 . . . ∂ αn n η, 0 ≤ |α| = α1+. . .+αn ≤ r) andL is a symmetric matrix with entries in A, under the differential constraintR η = 0. The optimal system is then defined by { η = Qξ, A ξ = 0, ∀ ξ ∈ F, with η = (ηl)1≤l≤p, π k = ∑ 1≤l≤p,0≤|β|≤r L k,l ∂ β ηl, A = Q̃ · B ·Q,