This paper discusses the asymptotic behavior of a sequence generated by non-linear recurrence relations, which arise in dynamic programming processes. The recurrence relations are of the form: $$ f_{N}(i)=\underset{\mathbf{q}}{Max}\left[b_{i}(\mathbf{q})+\sum_{i=1}^{M}a_{i i}(\mathbf{q})f_{N-1}(j)\right],\qquad N=1,2,\cdots, $$ with initial condition $ f_{0}(i)=c_{i} $. Although non-linear, these equations have quasi-linear properties that allow for a detailed analysis. The paper also considers similar recurrence relations involving integrals and difference equations. The main result is a theorem stating that under certain conditions, the sequence $ f_{N}(i) $ asymptotically behaves as $ N r $, where $ r $ is a scalar obtained by maximizing a certain expression over all possible values of $ \mathbf{q} $. The conditions on the functions $ b_{i}(\mathbf{q}) $ and $ a_{ij}(\mathbf{q}) $ ensure that the maximum is attained in the recurrence relations. The paper also discusses the properties of Markoff matrices and their asymptotic behavior, and provides a proof of the theorem. It concludes by discussing the implications of the results for dynamic programming processes and the importance of inter-linking in such systems. The paper references several related works and provides examples of processes of the general type discussed.This paper discusses the asymptotic behavior of a sequence generated by non-linear recurrence relations, which arise in dynamic programming processes. The recurrence relations are of the form: $$ f_{N}(i)=\underset{\mathbf{q}}{Max}\left[b_{i}(\mathbf{q})+\sum_{i=1}^{M}a_{i i}(\mathbf{q})f_{N-1}(j)\right],\qquad N=1,2,\cdots, $$ with initial condition $ f_{0}(i)=c_{i} $. Although non-linear, these equations have quasi-linear properties that allow for a detailed analysis. The paper also considers similar recurrence relations involving integrals and difference equations. The main result is a theorem stating that under certain conditions, the sequence $ f_{N}(i) $ asymptotically behaves as $ N r $, where $ r $ is a scalar obtained by maximizing a certain expression over all possible values of $ \mathbf{q} $. The conditions on the functions $ b_{i}(\mathbf{q}) $ and $ a_{ij}(\mathbf{q}) $ ensure that the maximum is attained in the recurrence relations. The paper also discusses the properties of Markoff matrices and their asymptotic behavior, and provides a proof of the theorem. It concludes by discussing the implications of the results for dynamic programming processes and the importance of inter-linking in such systems. The paper references several related works and provides examples of processes of the general type discussed.