The paper studies a PDE model for the growth of a tree stem or a vine, having the form of a differential inclusion with state constraints. The equations describe the elongation due to cell growth, and the response to gravity and to external obstacles.
The main theorem shows that the evolution problem is well posed, until a specific "breakdown configuration" is reached. A formula is proved, characterizing the reaction produced by unilateral constraints. At a.e. time $t$, this is determined by the minimization of an elastic energy functional under suitable constraints.
Citation: |
Figure 1. Left: at any point $\gamma(t,\sigma)$ along the stem, an infinitesimal change in curvature is produced as a response to gravity (or stems of other plants). The angular velocity is given by the vector $\omega(\sigma)$. This affects the position of all higher points along the stem. Right: At a given time $t$, the curve $\gamma(t,\cdot)$ is parameterized by $s\in [0,t]$. It is convenient to prolong this curve by adding a segment of length $T-t$ at its tip (dotted line, possibly entering inside the obstacle). This yields an evolution equation on a fixed functional space $H^2([0,T];\,\mathbb{R}^3)$.
Figure 2. Left: three configurations of the stem, relative to the obstacle. Right: in an abstract space, the first two configurations are represented by points $\gamma_1,\gamma_2$ on the boundary of the admissible set $S$ where the corresponding cones $\Gamma_1,\Gamma_2$ are transversal. On the other hand, $\gamma_3$ is a "breakdown configuration", satisfying all assumptions (26)-(27). Its corresponding cone $\Gamma_3$ is tangent to the boundary of the set $S$. Here the shaded region is the complement of $S$.
A. Bressan , M. Palladino and W. Shen , Growth models for tree stems and vines, J. Differential Equations, 263 (2017) , 2280-2316. doi: 10.1016/j.jde.2017.03.047. | |
L. Cesari, Optimization -Theory and Applications, Springer-Verlag, 1983. | |
G. Colombo and V. Goncharov , The sweeping processes without convexity, Set-Valued Anal., 7 (1999) , 357-374. doi: 10.1023/A:1008774529556. | |
G. Colombo and M. Monteiro Marques , Sweeping by a continuous prox-regular set, J. Differential Equations, 187 (2003) , 46-62. doi: 10.1016/S0022-0396(02)00021-9. | |
O. Leyser and S. Day, Mechanisms in Plant Development, Blackwell Publishing, 2003. | |
J. J. Moreau , Evolution problems associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977) , 347-374. doi: 10.1016/0022-0396(77)90085-7. | |
R. Rossi and U. Stefanelli , An order approach to a class of quasivariational sweeping processes, Adv. Diff. Equat., 10 (2005) , 527-552. | |
R. B. Vinter, Optimal Control, Birkhäuser, Boston, 2000. |
Left: at any point
For the two initial configurations on the left, the constrained growth equation (8) admits a unique solution. On the other hand, the two configurations on the right satisfy both (26) and (27) in (B). In such cases, the Cauchy problem is ill posed.
Left: three configurations of the stem, relative to the obstacle. Right: in an abstract space, the first two configurations are represented by points