It is well know from the mechanics of materials and structures that softness and slenderness re two key features that permit an elastic body to attain a large range of motion under moderate external forcing. The first feature, a material - i.e. local - property, is related to the small growth of the strain-energy density at a point as a function of the deformation tensor at the same point (see ciarlet_mathematical_elasticity). Typical examples of soft materials are polymers, such as rubbers and gels, in contrast to hard materials such as wood or steel. The second feature, a geometric property, is related to the degeneracy, for thin domains, of the best constants in the Korn-type inequalities which bound, in a suitable functional analytic framework, the displacement field in term of the deformation field (ciarlet mathematical elasticity). Thus, softness allows strain to be large with moderate involvement of energy, whereas slenderness allows displacement to be large with moderate involvement of strain.
\[ \left\{\begin{array}{l} -\vartheta^{\prime \prime}-\vec{h} \cdot D \vec{m}(\alpha+\vartheta)=0 \quad \text { in }(0,1) \\ \vartheta(0)=0 \\ \vartheta^{\prime}(1)=0 \end{array}\right.\label{eq:1} \]
where \[ D \vec{m}(v)=(-\sin v, \cos v), \quad \text { for all } v \in \mathbb{R} \] is the derivative of the function \(\vec{m}: \mathbb{R} \rightarrow \mathbb{R}^{2}\) defined by \(\vec{m}(v)=(\cos v, \sin v) ;\) moreover, such minimizer is unique if \[ |\vec{h}|<c_{p}^{-2} \] where \(c_{p}=2 / \pi\) is the best constant in the Poincaré-type inequality \[ \int_{0}^{1} v^{2} \leq c_{p}^{2} \int_{0}^{1}\left(v^{\prime}\right)^{2} \quad \text { for all } v \in C^{1}([0,1]) \text { such that } v(0)=0 \] Thus, if \(|\vec{h}|<c_{p}^{-2}\) then the state equation \(\left(P_{v}\right)\) defines a solution operator \[ B\left(0, c_{p}^{-2}\right) \times C([0,1]) \ni(\vec{h}, \alpha) \stackrel{\Theta}{\mapsto} \vartheta \in C^{2}((0,1)) \cap C^{1}([0,1]) \] which maps the control \((\vec{h}, \alpha)\) into the state \(\vartheta\).
The controllability of the rotation \(\vartheta\) is the basis for the application of magnetic actuation principle to shape-programmable soft matter. In this context, one can formulate the following:
\[ \bar{\vartheta}(0)=0, \quad \bar{\vartheta}^{\prime}(1)=0 \] find a control \((\vec{h}, \alpha)\) such that \(\eqref{eq:1}\) has unique solution \(\vartheta,\) and this solution is as close as possible to \(\bar{\vartheta}\).
It is not difficult to check that a solution exists regardless of the particular norm selected if there is a pair \((\vec{h}, \alpha)\) which nullifies the residual \(\{\mathrm{eq}: 9\}\) \[ r:=-\bar{\vartheta}^{\prime \prime}-\vec{h} \cdot \vec{m}(\alpha+\bar{\vartheta}) \] with \(\vec{h}\) satisfying \((1.6) .\) Precisely, assume that \[ M=\max_{s \in \mathbb{N}}\left|\bar{\vartheta}^{\prime \prime}(s)\right|<c_{p}^{-2} \] For \(\vec{h}=(M, 0), \quad \alpha(s)=\arcsin \left(\frac{\bar{\vartheta}^{\prime \prime}}{M}\right)-\bar{\vartheta}(s)\) we have \(r=0 .\) Thus \(\bar{\vartheta}\) is a solution of \(\eqref{eq:1}\) and this solution is unique, because \(\vec{h}\) satisfies the condition.