Global controllability and stabilization of the wave maps equation from a circle to a sphere

Abstract Continuing the investigations started in the recent work (Krieger and Xiang in Semi-global controllability of a geometric wave equation, 2022. arXiv:2205.00915 ) on semi-global controllability and stabilization of the $$(1+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional wave maps equation with spatial domain $$\mathbb {S}^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:math> and target $$\mathbb {S}^k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msup> </mml:math> , where semi-global refers to the $$2\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> -energy bound, we prove global exact controllability of the same system for $$k&gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and show that the $$2\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:math> -energy bound is a strict threshold for uniform asymptotic stabilization via continuous time-varying feedback laws indicating that the damping stabilization in Krieger and Xiang (2022) is sharp. Lastly, the global exact controllability for $$\mathbb {S}^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -target within minimum time is discussed.