Max-Plus Operator Semigroups

• Max-plus operator semigroups are families of max-plus linear operators on semimodules, using maximization and addition to mimic classical semigroup properties.
• They provide a robust framework for analyzing nonlinear evolution equations and dynamic programming with explicit kernel-based solution methods.
• Their applications span optimization, control theory, and Riccati equation analysis, offering novel spectral insights and efficient computational strategies.

A max-plus operator semigroup is a family of max-plus linear operators on a semimodule over the max-plus semiring, equipped with an algebraic structure mirroring classical semigroups, but with operations of maximization and addition replacing vector addition and scalar multiplication. Max-plus operator semigroups form the foundation for the analysis of nonlinear evolution equations, particularly in optimization, control, and Hamilton–Jacobi theory, and play a central role in recent advances in explicit solution formulas for Riccati-type equations via dynamic programming evolution operators (Fijavž et al., 2014, Zhang et al., 2014, Dower et al., 2014).

1. Algebraic and Analytical Foundations

Let $(\mathbb{R}^-,\oplus,\otimes)$ denote the max-plus semiring, where $\mathbb{R}^- = \mathbb{R} \cup \{ -\infty \}$, $a \oplus b = \max(a,b)$, and $a \otimes b = a + b$. The zero element is $-\infty$, and the unit is $0$. A semimodule $X$ over $R_{\max}$ is a set with internal operation $\oplus$ (max), external action $\otimes$ (addition), and the corresponding distributivities and idempotency (e.g., $a\otimes(x\oplus y) = a\otimes x \oplus a\otimes y$).

A max-plus linear operator $T:X\to X$ satisfies

$$T(a\otimes f \oplus b\otimes g) = a\otimes T(f) \oplus b\otimes T(g)$$

for all $f,g \in X$, $a,b \in \mathbb{R}^-$, and $T(-\infty) = -\infty$.

A family $\{T(t)\}_{t\geq 0}$ is a (strongly continuous) max-plus operator semigroup if

• $T(0)=\mathrm{Id}$,
• $T(s+t) = T(s) \circ T(t)$ for all $s, t\geq 0$,
• $T(t)$ is max-plus linear for each $t$,
• $T(t)f\to f$ in norm as $t\to 0^+$ for each $f$.

2. Generators and Resolvents

The generator $A$ of a strongly continuous max-plus semigroup $\{T(t)\}$ is defined by

$Af := \lim_{h\to 0^+} \frac{T(h)f - f}{h}$

when the limit exists. The generator is itself max-plus linear on its domain.

The max-plus resolvent operator for $\lambda > 0$ is defined via the supremal formula

$$R(\lambda)f = \sup_{t\geq 0} [T(t)f \oplus (-\lambda t)] = \sup_{t \geq 0} \left(T(t)f - \lambda t\right).$$

This operator satisfies the identity $R(\lambda)R(\mu) = R(\min\{\lambda,\mu\})$ and a max-plus Yosida-type estimate $\|\lambda \otimes R(\lambda)f\| \leq \|f\|$ (Fijavž et al., 2014).

A Hille–Yosida-type theorem holds: If $A$ is max-plus linear and the resolvent $R(\lambda)$ is well-defined with certain norm and monotonicity estimates, then there exists a unique strongly continuous max-plus semigroup generated by $A$.

3. Kernel Representations and Fundamental Solution Semigroups

In discrete-time dynamic programming settings, with $S: F \to F$ as the one-step evolution operator,

$$(S\phi)(x) = \sup_{w} \left\{ \tfrac{1}{2} x^\top \Phi x - \tfrac{\gamma^2}{2} |w|^2 + \phi(Ax + Bw) \right\},$$

the $k$-step operators $S_k = S\circ \cdots \circ S$ ($k$-fold) form a max-plus semigroup. Each $S_k$ acts as a max-plus integral operator with kernel $G_k(x,y)$: $$(S_k \phi)(x) = \int_{\mathbb{R}^n}^{\oplus} G_k(x,y) \otimes \phi(y) \, dy$$ where $\int^{\oplus}$ denotes the supremal integral.

Two explicit semigroup constructions emerge for quadratic kernels, distinguished as:

• Dual-space semigroup: via Legendre–Fenchel transform–based duality, yielding max-plus integral operators with quadratic kernels $B_k(y,z) = \tfrac{1}{2} [y;z]^\top \Theta_k [y;z]$. The semigroup property reads $B_{k+\ell} = B_k \circ B_\ell$, with kernel convolution and Hessian composition following a blockwise Schur complement.
• Primal-space semigroup: defined by kernels $$S_k(x,y) = \tfrac{1}{2} [x; y]^\top \Lambda_k [x; y]$$, with the same semigroup and convolution structure. The two kernel families $\{\Theta_k\}$ and $\{\Lambda_k\}$ are related via explicit isomorphisms involving flipping and Schur-complement operations (Zhang et al., 2014).

Both semigroups precompute kernel families, allowing closed-form propagation of Riccati-iterates for arbitrary initial quadratic terms.

4. Semigroups in Operator Differential Riccati Theory

For infinite-dimensional operator-valued Riccati equations

$$\dot P(t) = A'P(t) + P(t)A + P(t)\sigma\sigma'P(t) + C$$

the max-plus operator semigroup arises from embedding the Riccati flow into a finite-horizon optimal control problem. By exploiting the semiconvexity of the value function and the max-plus linearity of the discrete evolution $S_t$, one constructs a time-indexed family of max-plus integral operators $\{\mathcal{B}_t^\oplus\}$ in a dual space. These act via quadratic kernels and satisfy

$$\mathcal{B}^{\oplus}_{t+s} = \mathcal{B}^{\oplus}_t \circ \mathcal{B}^{\oplus}_s$$

with explicit kernel convolution. The duality is realized through the Legendre–Fenchel transform with quadratic kernels, admitting a general, closed-form propagation scheme for Riccati solutions initialized in large semiconvex/semiconcave classes. For any $P_0$ in this class, $P(t)$ is expressed as a pseudo-inverse (Moore–Penrose) formula in terms of precomputed kernel blocks (Dower et al., 2014).

5. Canonical Examples and Applications

A canonical example is the Lax–Oleinik semigroup associated with Hamilton–Jacobi equations. For a convex Hamiltonian $H$ and Tonelli Lagrangian $L$, the semigroup is defined by

$$T(t)f(x) = \sup_{\gamma,\,\gamma(t)=x} \left[ f(\gamma(0)) - \int_0^t L(\gamma(s),\dot\gamma(s)) ds \right]$$

The generator acts as $A f(x) = H(x, \nabla f(x))$. This construction gives viscosity solutions to Hamilton–Jacobi equations and exemplifies the max-plus semigroup’s ability to capture nonlinear evolution in a linear-operator-theoretic framework (Fijavž et al., 2014).

In control theory, max-plus operator semigroups enable the direct construction of fundamental solutions for Riccati and operator-DREs. Precomputed quadratically-parameterized kernel families provide efficient, non-iterative propagation of solutions after an initial kernel assembly step, supporting both finite and infinite-dimensional computational practice (Zhang et al., 2014, Dower et al., 2014).

6. Spectral Theory, Ergodic Limits, and Structure

Max-plus semigroups admit a spectral theory in terms of additive eigenvalues and eigenfunctions: $T(t)u = \lambda t \oplus u, \quad \forall t \ge 0$ The ergodic theorem ensures, under compactness and regularity conditions, convergence of the form

$\lim_{t \to \infty} (T(t)f - \Lambda t) = v$

with a corresponding eigenrelation $T(t) v = \Lambda t \oplus v$. These results parallel the classical Perron–Frobenius theory, and ensure asymptotic rigidity and stability in both finite- and infinite-dimensional contexts (Fijavž et al., 2014).

Spectral properties reflect the long-time growth rates of dynamical and control systems modeled via max-plus operator semigroups. In the dual space semigroup for Riccati equations, the dominant eigenvalue corresponds to the “growth rate” of cost over the horizon and matches classical interpretations.

7. Connections, Extensions, and Structural Aspects

Max-plus operator semigroups generalize classical (linear) semigroup theory by replacing vector addition and scalar multiplication with the operations of maximization and addition. The Yosida approximation in this setting involves compositions of the form $(I \oplus \tfrac{t}{n}A)^{\otimes n}$ rather than the classical $(I + \tfrac{t}{n}A)^n$. The semigroup and generator framework, resolvent identity, existence/uniqueness, continuous dependence on data, and spectral theory persist under these generalized operations.

Further, results on max-plus matrix semigroups—such as generating sets and relations for the monoid of $2\times2$ matrices over finite max-plus semirings—illuminate concrete structural properties and presentations that serve as finitary analogues of operator semigroups. These finite presentations inform the computational structure and combinatorics underlying max-plus operator semigroups, especially in the context of tropical and idempotent analysis (East et al., 2020).

Max-plus operator semigroups form a unifying structure underpinning a range of nonlinear, dynamic, and optimization-theoretic phenomena, linking discrete, finite, and infinite-dimensional systems through their kernel-based, sup-convolutional, and spectral properties.