A bounded endpoint selector yields small nonlinear exact steering

Assume the Axiom of Choice and fix $T>0$. Let $E$, $U$ and $X$ be real Banach spaces, meaning complete real normed vector spaces. Write $C_T(E)=C([0,T];E)$ for norm-continuous paths with the supremum norm, and similarly for $U$. Suppose $X$ is continuously embedded in $C_T(E)$, with ${\sup }_{0\le t\le T}\mathrm{\parallel }v(t){\mathrm{\parallel }}_E\le e\mathrm{\parallel }v{\mathrm{\parallel }}_X$ for some $e\ge 0$. Let $A:E\to X$ and $D_c:C_T(U)\to X$ be bounded real-linear maps. The associated controlled linear solution is $Ax+D_{c}h$; the same maps are used throughout this statement. Suppose a bounded real-linear selector $G:E\times E\to C_T(U)$ satisfies $\mathrm{\parallel }G(x,y){\mathrm{\parallel }}_{C_T(U)}\le g(\mathrm{\parallel }x{\mathrm{\parallel }}_E+\mathrm{\parallel }y{\mathrm{\parallel }}_E)$ and that $Ax+D_{c}G(x,y)$ has initial value $x$ and terminal value $y$ for every $x,y\in E$.

Let $m\in \mathbb{R}$ satisfy $m>1$, and let $C\ge 0$. Powers of nonnegative norms use $0^{\alpha }=0$ for real $\alpha >0$. Suppose $N:X\to X$ satisfies $N(0)=0$, $N(v)(0)=0$ for every $v\in X$, $\mathrm{\parallel }N(v){\mathrm{\parallel }}_X\le C(\mathrm{\parallel }v{\mathrm{\parallel }}_X^2+\mathrm{\parallel }v{\mathrm{\parallel }}_X^m)$, and $\mathrm{\parallel }N(v)-N(u){\mathrm{\parallel }}_X\le C(\mathrm{\parallel }u{\mathrm{\parallel }}_X+\mathrm{\parallel }v{\mathrm{\parallel }}_X+\mathrm{\parallel }u{\mathrm{\parallel }}_X^{m-1}+\mathrm{\parallel }v{\mathrm{\parallel }}_X^{m-1})\mathrm{\parallel }v-u{\mathrm{\parallel }}_X$ for all $u,v\in X$. For endpoints $x_0,x_T\in E$, define $h_v=G(x_0,x_T-N(v)(T))$ and $\mathcal{T}(v)=A{x}_0+N(v)+D_c{h}_v$.

There are $r>0$ and $\delta >0$, depending only on the displayed constants and maps for the fixed time interval, such that whenever $\mathrm{\parallel }{x}_0{\mathrm{\parallel }}_E+\mathrm{\parallel }{x}_T{\mathrm{\parallel }}_E\le \delta$, the map $\mathcal{T}$ takes the closed radius-$r$ ball of $X$ into itself and is a contraction there with constant at most $1\mathrm{/}2$. Its unique fixed point in that ball has initial value $x_0$ and terminal value $x_T$. Its selected control is $h_v\in C_T(U)$. No uniqueness outside that ball is asserted.

Continuous and mixed time-space function spaces

Reachability and observation vocabulary for controlled evolutions

Banach space

The Axiom of Choice

Endpoint maps and exact steering from a null-control operator

A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point

A bounded linear operator between normed spaces

The operator norm as the least bound and as the unit-sphere or unit-ball supremum

Real powers for positive bases, with the zero-base positive-exponent convention

The argument also applies to complex Banach spaces regarded as real spaces, provided all maps satisfy the stated real-linear hypotheses. By Realification preserves a complex norm and its metric, realification preserves the norm and its metric; consequently the Cauchy sequences and their limits are unchanged, so completeness is preserved. In an evolution with a bounded real-linear forcing map $D:L^1(0,T;E)\to X$ and a bounded real-linear control-to-forcing map $B:C_T(U)\to L^1(0,T;E)$, one may take $D_c=DB$ after verifying the endpoint identity for that same composition. The composition is bounded and real-linear, with $\mathrm{\parallel }DB\mathrm{\parallel }\le \mathrm{\parallel }D\mathrm{\parallel }\mathrm{\parallel }B\mathrm{\parallel }$, by Composition satisfies ‖ST‖≤‖S‖ ‖T‖. For a real-linear Schrödinger evolution, any factor $-i$ belongs in $B$; no commutation with the evolution is used.