Gradient Structures Associated with a Polynomial Differential Equation

: In this paper, by using the characteristic system method, the kernel of a polynomial differential equation involving a derivation in R n is described by solving the Cauchy Problem for the corresponding ﬁrst order system of PDEs. Moreover, the kernel representation has a special signiﬁcance on the space of solutions to the corresponding system of PDEs. As very important applications, it has been established that the mathematical framework developed in this work can be used for the study of some second-order PDEs involving a ﬁnite set of derivations.


Introduction
Throughout time, gradient-type representations for some solutions, gradient systems in a Lie algebra and the algebraic representation of gradient systems, have been investigated, with remarkable results, by Vârsan [1]. Moreover, stochastic partial differential equations (SPDEs) of Hamilton-Jacobi type including non F T -adapted solutions have been studied in Ijacu and Vârsan [2]. By using the commuting property of the drift and diffusion vector fields with respect to the usual Lie bracket, a representation for a classical solution of some nonlinear SPDEs driven by Fisk-Stratonovich stochastic integral was constructed by Iftimie et al. [3]. Furthermore, sufficient conditions for linear subspaces of smooth vector fields in order to be written as a kernel of some linear first order partial differential equations are have been formulated and proved in Parveen and Akram [4]. Further, Treanţȃ and Vârsan [5] proved that solutions associated with an extended affine control system can be obtained as a limit process using solutions for a parameterized affine control system and weak small controls. Recently, Treanţȃ [6] studied affine control systems with jumps for which the ideal generated by the drift vector field can be imbedded as a kernel of a linear first-order partial differential equation. Mainly, these references motivate the present study. For other different but connected viewpoints regarding this subject, the reader is directed to Friedman [7], Sussmann [8], Crandall and Souganidis [9], Sontag [10], Bressan and Shen [11], Nonlaopon [12], Saira et al. [13] and Treanţȃ [14][15][16].
In this paper, taking into account the results included in the quite recently work Treanţȃ [17] (the kernel of a polynomial of scalar derivations is described by solving Cauchy Problems for the corresponding system of ODEs; also, a gradient representation for the associated Cauchy Problem solution is derived), we investigate the kernel of a polynomial differential equation involving a derivation in R n by solving the Cauchy Problem for the corresponding first order system of PDEs. Furthermore, we extend a solution by considering Radon measures and their bounded variation functions, or Wiener and Levy processes. Moreover, it is established that the kernel representation has a special significance on the space of solutions to the corresponding system of PDEs. This paper is organized as follows. In Section 2, in order to delineate certain steps in the solving algorithm proposed for the main result (Theorem 1), some preliminary results are formulated. More precisely, two crucial lemmas for the present paper are mentioned. Further, we establish two important remarks. A gradient structure for the associated Cauchy Problem solution is provided by Remark 1. The final part of this section, including Remark 2, extends a solution considering Radon measures and their bounded variation functions, or using Wiener (or Levy) processes. The aim of Section 3 is to provide a characterization for the kernel of a polynomial differential equation involving a derivation in R n . Specifically, through the use of the characteristic system method and some results formulated in Section 2, the associated Cauchy Problem solution is derived (see Theorem 1). Moreover, this solution has a special significance on the space of solutions to the corresponding first order system of PDEs. Finally, Section 4 concludes the paper.

Preliminary Results
In this section, taking into account a very recent work (see Treanţȃ [17]), some auxiliary results are formulated.
Let 0 ∈ I ⊆ R be an open interval. Consider a polynomial of the scalar derivation d dt , where m ≥ 1, a j ∈ L ∞ (I) , j ∈ {1, 2, ..., m}. Define and consider Ker (P m ) ⊆ H m ∞ (I), where The procedure of characteristic systems (see Friedman [7], Vârsan [1]) allows us to describe Ker (P m ) by solving Cauchy Problems for the corresponding system of ODEs using a vector variable y = col (y 1 , y 2 , ..., y m ) , Here, the (m × m) constant matrices A and B i , i = 1, m, are defined by where {e 1 , ..., e m } is the canonical basis and 0 ∈ R m is the origin. By definition and making a direct computation, we get with O-null matrix, and The Cauchy Problem solution for (4) is represented by where {ŷ(t; y 0 ) : t ∈ I} fulfils the following linear system (initial value problem) Write the (m × m) matrices as follows where the linear mapping ad A : ). In addition, using (7), (8) and (12), we get Denote N = m 2 and define N matrices {C 1 , C 2 , ..., C N } ⊆ M m×m , as follows With these notations, we write ODE (10) as follows where Y j (y) := C j y, j ∈ {1, 2, ..., N}.

Lemma 1 ([17]
). Consider {C 1 , C 2 , ..., C N } defined in (14), with N = m 2 . Then {C 1 , C 2 , ..., C N } is a basis for M m×m and .., e m } ⊆ R m is the canonical basis. Then ∀y 0 ∈ S i is a stationary point f or ODE (10). (17) In addition, for each y 0 ∈ S ⊥ i , y 0 = 0, the following statements are valid: In the following, we establish two important remarks for the main result associated with this paper.

Main Results
This section contains the main result associated with the present paper. In order to formulate and prove it, we start with the following mathematical tools and hypotheses.
Proof. Taking into account the aforementioned computations (see relations (53) and (54)), the proof is immediate and complete.

Conclusions and Further Developments
In this paper, we have investigated the kernel of a polynomial differential equation involving a derivation in R n by solving the Cauchy Problem for the corresponding first order system of PDEs. Moreover, we have proved that the kernel representation has a special significance on the space of solutions to corresponding system of PDEs.
The mathematical framework developed in this work can be easily extended for the study of some higher-order hyperbolic, parabolic or Hamilton-Jacobi equations involving a finite set of derivations.
Funding: The APC was funded by University Politehnica of Bucharest, "PubArt" program.