# Dark Type Dynamical Systems: The Integrability Algorithm and Applications +
## 2 Nonlinear Dynamical Systems and the Lax Type Representation
### An Optimal Control Problem Aspect  
Consider now a smooth nonlinear dynamical system  
\[v_{t}=K[v,u] \tag{46}\]  
on a \(2\pi\)-periodic functional manifold \(M_{\flat}\subset C(\mathbb{R}/\left\{2\pi\mathbb{Z}\right\};\mathbb{R}^{m(p)})\) and depending parametrically on a \(2\pi\)-periodic functional variable \(u\in M_{u}\subset C(\mathbb{R}/\left\{2\pi\mathbb{Z}\right\};\mathbb{R}^{m(u)})\) and pose the following Bellman-Pontriagin type optimal control problem [35; 36] on a temporal interval \([0,T]\subset\mathbb{R}_{+}\):  
\[\operatorname*{arg\,inf}_{(v,u)\in M_{\flat}\times M_{u}}\int_{0}^{T}dt\int_{ 0}^{2\pi}\mathcal{L}[v,u]dx=\;?, \tag{47}\]  
under condition that the evolution flow (46) possesses a smooth conserved quantity \(\gamma=\int_{0}^{2\pi}\gamma[u,v]dx\in\mathcal{D}(M_{\flat}\times M_{u})\), that is \(d\gamma/dt=0\) on the combined manifold \(M_{\flat}\times M_{u}\) for all \(t\in[0,T]\). The latter, in particular, means that we need to determine such an additional evolution flow  
\[u_{t}=F[v,u] \tag{48}\]  
on the control manifold \(M_{\flat}\), which ensures the existence of the mentioned above conserved quantity \(\gamma\in\mathcal{D}(M_{\flat}\times M_{u})\). The problem above is solved by means of construction of the extended Lagrangian functional  
\[\mathcal{L}_{\mu,\vartheta}:=\int_{0}^{T}dt\left[\int_{0}^{2\pi}\mathcal{L}[v, u]dx+\mu(t)\frac{d}{dt}\int_{0}^{2\pi}\gamma[u,v]dx+(\psi|v_{t}-K[v,u])\right] \tag{49}\]by means of the Lagrangian multipliers \(\mu\in C^{1}_{0}([0,T];\mathbb{R})\) and \(\psi\in\ C^{1}_{0}([0,T];T^{*}(M_{\varphi}))\) almost everywhere with respect to the temporal parameter \(t\in[0,T]\), and finding its critical points:  
\[\mathcal{L}^{{}^{\prime}*}_{\varphi,\mu,\varphi}(1)=0\sim\text{grad}_{v}[ \mathcal{L}\ -\dot{\mu}(t)\gamma][v,u]-(\psi_{t}+K^{{}^{\prime}*}_{\varphi}[v,u]\psi)=0 \tag{50}\]  
for all \((v,u)\in M_{\mathbb{D}}\times M_{u}\) jointly with the condition that \(d\gamma/dt=0\). The obtained functional relationship (50) under the condition \(\mathcal{L}\ =\dot{\mu}(t)\gamma\) for \(t\in[0,T]\) reduces to the following Noether-Lax condition  
\[\psi_{t}+K^{{}^{\prime}*}_{\varphi}[v,u]\psi=0 \tag{51}\]  
on the Lagrangian multiplier \(\psi\in\ C^{1}_{0}([0,T];T^{*}(M_{\mathbb{D}}))\). The functional relationship (51) presents an evolution partial differential equation on the vector-function \(\psi\in\ C^{1}_{0}([0,T]\times\mathbb{R};\mathbb{R})\otimes\mathbb{C}\), which always possesses [37] an asymptotic as \(\lambda\to\infty\) solution in the following form:  
\[\psi=(1,a_{1}(x;\lambda),a_{2}(x;\lambda),...,a_{m(v)-1}(x;\lambda))^{\intercal }\exp[\lambda^{s(\psi)}t+Z(x;\lambda)] \tag{52}\]  
for some fixed \(s(\psi)\in\mathbb{Z}_{+}\), where \(z(x;\lambda)\sim\sum_{j\in\mathbb{Z}_{+}}z_{j}(x)\lambda^{-j+s(Z)},s(Z)\in \mathbb{Z}_{+}\), and whose coefficients \(z_{j}\in C^{\infty}(\mathbb{R};\,\mathbb{R}),j\in\mathbb{Z}_{+}\), can be easily calculated by means of the respectively derived recurrent differential-functional relationships, depending exclusively on the derivatives \(\partial z_{j}/\partial t\), \(\partial z_{j}/\partial x\), \(j\in\mathbb{Z}_{+}\). If, in addition, the coefficients \(z_{j}\in C^{\infty}(J(\mathbb{R}/\{2\pi\mathbb{Z}\};\mathbb{R}^{m(v)}\times \mathbb{R}^{m(u)});\mathbb{R}),j\in\mathbb{Z}_{+}\), then _there exist a priori_ local quantities \(\partial z_{j}/\partial x:=\sigma_{j}[v,u]\in C^{\infty}(J(\mathbb{R}/\{2\pi \mathbb{Z}\};\mathbb{R}^{m(v)}\times\mathbb{R}^{m(u)});\mathbb{R}),j\in \mathbb{Z}_{+}\), such that \(\partial\sigma_{j}[v,u]/\partial t=\partial\big{(}\partial z_{j}/\partial t \big{)}/\partial x\), simply meaning that the quantities \(\gamma_{j}:=\int_{0}^{2\pi}\sigma_{j}[u,v]dx,j\in\mathbb{Z}_{+}\), are conserved, that is \(\frac{d}{dt}\int_{0}^{2\pi}\sigma_{j}[u,v]dx=0\) for all \(j\in\mathbb{Z}_{+}\).