Elastic Problems and Optimal Control: Integrable Systems

Velimir Jurdjević

This paper, loosely described as the variations on the Euler-Kirchhoff elastic theme, focuses on the class of variational problems on an orthonormal frame bundle of a Riemannian space of constant curvature and introduces optimal control theory as an important ingredient for their solutions. The fundamental spaces are the Euclidean space, the sphere and the hyperboloid, and their orthonormal frame bundles coincide with the isometry groups $\operatorname{SE}_n(R)$, $\operatorname{SO}_{n+1}(R)$ and $\operatorname{SO}(1,n)$. In each of these cases, the underlying manifold $M_\epsilon$ with its curvature $\epsilon=0,\pm 1$ can be realized as the quotient $M_ep=G_\epsilon/K$, where $G_\epsilon$ denotes the appropriate isometry group and where $K=\operatorname{SO}_n(R)$. The pair $(G_\epsilon,K)$ induces a Cartan decomposition $\mathfrak g_\epsilon=\fp_\epsilon+\mathfrak k$ of the Lie algebra $\mathfrak g_\epsilon$ of $G_\epsilon$, where $\mathfrak k$ is the Lie algebra of $K$ and where $\mathfrak p_\epsilon$ is the orthogonal complement of $\mathfrak k$ relative to the Cartan-Killing form on $\mathfrak g_\epsilon$. Kirchhoff's formalism used to model the equilibrium configurations of a thin elastic rod subject to bending and twisting torques at its ends admits natural formulation on these groups as an optimal control problem of optimizing the energy integral $\frac12\int_0^T\langle u(t),Qu(t)\rangle\,dt$ over the trajectories of the control system $\frac{dg}{dt}=g(t)(A+u(t))$ that satisfy fixed boundary conditions in $G_\epsilon$. Here, $A$ a fixed element in $\mathfrak p_\epsilon$, $u(t)$ is an arbitrary curve in $\mathfrak k$, $Q$ is a positive definite $n\times n$ matrix and $\langle X,Y\rangle=-\frac12peratorname{Tr}(XY)$, $X,Yı\mathfrak k$. The paper then singles out the integrable cases of the Hamiltonians associated with these optimal problems obtained by the Maximum Principle. The paper also defines a symplectic structure over quasi-periodiic curves on three dimensional spaces of constant curvature and shows that the Heisenberg's magnetic equation corresponds to the Hamiltonian flow associated with $\frac12\int_0^T\kappa^2(s)\,ds$ over such curves with $\kappa$ equal to the curvature of the curve. Finally, the paper gives the exact correspondence between the Heisenberg's magnetic equation and the nonlinear Schroedinger's equation and relates the soliton solutions to the elastic curves.