As　a　heuristic　algorithm　to　solve　the　nonlinear　optimal　control　problem,　adaptive　dynamic　programming　is　commonly　constructed　from　one-step　temporal　difference　learning.　Eligibility　trace　can　effectively　speed　up　controller　learning　by　considering　the　multi-step　information　in　reinforcement　learning.　However,　eligibility　trace　will　bring　in　additional　computational　consumption　and　increase　the　learning　burden.　This　paper　attempts　to　take　advantage　of　eligibility　trace　and　avoids　the　high　learning　consumption　at　the　same　time.　Therefore,　an　event-based　neural　dynamic　programming　(A)　[ENDP(A)]　algorithm　via　the　actor-critic　framework　is　constructed　to　solve　the　near-optimal　control　problem　of　unknown　discrete-time　systems.　First,　the　modified　forward　view　with　eligibility　trace　is　derived,　which　is　suitable　for　engineering　practice.　Second,　based　on　the　event-triggered　mechanism,　ENDP(A)　is　designed　to　relieve　the　pressure　of　communication　consumption.　Then,　the　event-based　system　is　proven　to　ensure　the　input-to-state　stability　under　a　suitable　triggering　condition.　Moreover,　three　neural　networks　are　given　to　approximate　the　one-step　cost　function,　the　n-step　cost　function,　and　the　control　law,　respectively.　Finally,　two　typical　experimental　simulation　examples　are　presented　to　verify　the　effectiveness　of　the　ENDP(A)　algorithm.