Agentic artificial intelligence systems, capable of autonomous perception, planning, tool use, and multi-step action, are increasingly proposed for critical engineering domains where decisions carry physical, operational, or economic consequences. This survey addresses a gap in current literature by treating trustworthiness, whether agentic behavior can be verified, audited, and trusted under the constraints that engineering practice actually requires, as a first-class engineering property, rather than evaluating agentic AI by task capability alone. The study adopts a trustworthiness model organized around five cross-cutting dimensions: safety and constraint satisfaction; robustness and reliability; transparency and interpretability; accountability and auditability; and privacy and security. This is mapped onto an agentic assurance workflow spanning perception through audit. Building on this foundation, agentic systems architectures, threats, concrete trust mechanisms, and quantitative metrics are surveyed for direct application in agentic systems development and evaluation. These principles are then examined across four constraint-bound engineering domains: power systems, autonomous vehicles/robotics/UAVs, high-performance computing, and communication networks, identifying recurring design patterns, shared failure modes, and domain-specific gaps. Synthesizing across those domains, agentic AI trustworthiness is shown to be a single problem, with a path outlined toward a reusable, cross-domain assurance framework analogous to the graded certification regimes used by mature safety-critical engineering fields.

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