## 2 The Problem of Thermoelasticity  
The one-dimensional thermoelasticity problem is considered in [3; 44]. It represents the most dangerous scenario in terms of the intensity of the temperature shock impact on the movement of the small spacecraft [51]. This is enough to assess the need to take into account the temperature shock. However, for the development of a control algorithm, where various scenarios should be provided, such a statement is not enough. Therefore, in this paper, we use the two-dimensional formulation of the thermoelasticity problem described in [25; 34; 52]. It is different from the one-dimensional one because the initial deformed state of the solar panel at the time of the temperature shock is taken into account. At the same time, the scenario for implementing the one-dimensional task is also included in this approach as a special case. Figure 1 shows a scheme of the symmetrical small spacecraft for the temperature shock studies. The dotted line in Figure 1 indicates the deformed initial position of the flexible element. The solid line shows its undeformed flat position. Let us assume that the initial deflections of both elastic elements coincide and all simplifying assumptions for the symmetric formulation of the thermoelasticity problem described in [3] are valid. Mathematically, the two-dimensional thermoelasticity problem is represented by the following system of equations [25; 34; 52]:  
\[\left\{ \begin{array}{l}\frac{\partial T(x,z,\ t)}{\partial t}=a^{2}\Big{(} \frac{\partial^{2}T(x,z,\ t)}{\partial x^{2}}+\frac{\partial^{2}T(x,z,\ t)}{ \partial z^{2}}\Big{)},\ \ 0\leq x\leq l,\ 0\leq z\leq h,\ t>0;\\ \frac{\partial^{4}u_{z}(x,\ t)}{\partial x^{4}}+\rho\,h\frac{\partial^{2}u_{ z}(x,\ t)}{\partial t^{2}}=-2\mu\alpha\int\limits_{0}^{h}\big{[}2\frac{\partial T(x,z,\ t)}{\partial z}+2\frac{\partial^{2}T(x,z,\ t)}{ \partial z^{2}}\big{]}dz+\\ +\frac{\partial^{2}u_{z}(x,\ t)}{\partial x^{2}}\,\sigma_{xz}(x,\ t),\ 0\leq x\leq l,\ z=\frac{h}{t},\ t>0;\\ \Big{(}\lambda\frac{\partial T(x,z,\ t)}{\partial n}\Big{)}=Q-\varepsilon \sigma\big{(}T^{4}(x,h,\ t)-T_{C}^{4}\big{)},\ 0\leq x\leq l,\ z=h,\ t>0;\\ \Big{(}\lambda\frac{\partial T(x,z,\ t)}{\partial n}\Big{)}=-\varepsilon \sigma\big{(}T^{4}(x,\,0,\ t)-T_{C}^{4}\big{)},\ 0\leq x\leq l,\ z=0,\ t>0;\\ T(\ x,\ z,\ 0)=T_{0}=const,\ 0\leq x\leq l,\ 0\leq z\leq h,\ t=0;\\ u_{z}(0,\ t)=0,\ x=0,\ t>0;\\ \frac{\partial u_{z}(x,\ t)}{\partial x}=0,\ x=0,\ t>0;\\ \frac{\partial^{2}u_{z}(x,\ t)}{\partial x^{2}}=0,\ x=l,\ t>0;\\ \frac{\partial^{3}u_{z}(x,\ t)}{\partial x^{3}}=0,\ x=l,\ t>0;\\ u_{z}(x,\ 0)=u_{z}(x,\ 0),\ 0\leq x\leq l,\ \ t=0.\end{array} \right. \tag{1}\]  
where \(a\) is the coefficient of temperature conductivity; \(h\) is the thickness of the flexible element; \(u_{z}=u_{z}(x,\ t)\)--the deflection of the flexible element points in the direction of the \(z\)-axis; \(u_{z0}=u_{z0}(x,\ 0)\)--the initial deflection of the flexible element points in the direction of the \(z\)-axis; \(\rho\) is the density of the flexible element; \(D\) is the cylindrical rigidity of the flexible element for bending; \(l\) is the length of the flexible element; \(\lambda\) is the coefficient of thermal conductivity; \(\varepsilon\) is the degree of blackness of the flexible element material; \(\sigma\) is the Stefan-Boltzmann constant; \(n\) is the unit vector of the normal to the surface element; and \(\sigma_{xz}\) is the component of the stress tensor. The first equation of System (1) is a classical equation of two-dimensional thermal conductivity when the temperature field has the following form: \(T=T(x,\ z,\ t)\). At the same time, [PERSON]'s law is valid. Figure 1: The considered small spacecraft scheme. In the second equation of System (1), it is assumed that the deflection in the framework of the two-dimensional problem can be represented as \(u_{z}=u_{z}(x,\ t)\)[53]. It is a [PERSON] equation for thin plates [54; 52]. Its satisfaction is required only in the middle of the plate within the framework of this statement.