4 ��Let ��ij = ��, ��ij = ��, and Fi = F be real constants. If there exists a nonnegative solution f of the controlled kinetic framework compound libraries (16) such that fi(t, u) = 0 as u ?Du, then the 1th-order moment 1[f](t) is solution of the following Riccati nonlinear ordinary differential ?��(1?��)��(t)?1[f](t).(21)Proof ��The?equation:ddt?1[f](t)=F(�̡�(t)?(?1[f](t))2) interaction operator [fi, fj] can be written as ?�ɦ�j(t)fi(t,u)+�ɦЦ�j(t)fi(t,u).(22)Multiplying?follows:?ij[fi,fj](t,u)=?ij[fi,fj](t,u) both sides of ij[fi, fj] by u and integrating over Du, we =?�ɦ�j(t)(1?��)��Duufi(t,u)du.(23)Summing with???have��Duu?ij[fi,fj](t,u)du respect to j, multiplying by vi, and summing with respect to i, we =?�ɦ�(t)(1?��)?1[f](t).

(24)Multiplying??obtain��i=1nvi��j=1n��Duu?ij[fi,fj](t,u)du by u and vi the second term of the left hand side of (16), integrating with respect to the activity variable, performing integration by parts and summing with respect to i, we =(?1[f](t))2?�̡�(t)(25)and then the??have��i=1nvi��Duu?u((1?u?1[f](t))fi(t,u))du proof. According to Theorem 4, the solution of the Riccati equation (21) can be obtained as follows. The Riccati equation ��3(t)=?F??�̡�(t),(27)if??��2(t)=��(1?��)��(t),??+��(1?��)��(t)?1[f](t)?F�̡�(t)=0.(26)Setting��1(t)=F,??readsddt?1[f](t)+F(?1[f](t))2 ?1[f](t)�� is a solution of (26), the general integral can be written as?1[f](t)=?1[f](t)��+1��(t),(28)where ��(t) is solution of�ˡ�?[��2(t)+2��1(t)?1[f](t)��]��=��1(t).(29)A nonnegative and constant solution of (26) is?1[f](t)��=��2(1?��)2��2(t)+4F2�̡�(t)?��(1?��)��(t)2F.

(30)Therefore, the solution of (26) can be written as follows:?1[f](t)=?1[f](t)��+e��0t��(��)d��(?1[f](0)??1[f](t)��)?1+F��0te��(��)d��,(31)where��(��)=��2(1?��)2��2(��)+4F2�̡�(��).(32)The next theorem gives the evolution equation for all moments where p is an odd number.Theorem 5 ��Let p , q be an odd number and t �� 0. Then, the (p, q)th-order moment of the distribution function f satisfies the following Riccati nonlinear ordinary differential +�ɦ�(t)(1?��)?p,q[f](t),(33)where��~(t)=��i=1nvip��i(t).(34)Moreover,?equation:ddt?p,q[f](t)=pF?p,q?1[f](t)(��~(t)??p,q[f](t)) ifp,q[f](t) is initially bounded, it remains bounded for all t > 0. Proof ��The proof follows by multiplying both sides of (14) by up and performing integration by parts on the control term. 4.

Research Perspectives The controlled kinetic framework proposed in this paper allows the derivation of specific models for multicellular systems characterized by nonconservative interactions. This framework belongs to the class of thermostated kinetic for active particles Batimastat models.The mathematical framework (16) can be further generalized in order to include the role of mutations; see Nowak [26]. This is an important issue in the cancer modeling [27, 28]. A future research perspective is the generalization of the mathematical framework (16) to open systems subjected to external actions at the microscopic scale, for example, the role that the